Riemann Sums in Summation Notation

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arbitrary function. These formulations help us define the definite integral. The basic idea behind these sums is to divide the area that is supposed to be calculated into small rectangles and calculate the sum of their areas. These areas are not accurate, but they help a lot in getting an idea about the actual area. The more the number of rectangles, the closer it takes us to the actual area. Let’s look at Riemann sums with sigma notations. </p> <h3>Riemann Sums</h3> <p>Definite integrals are an important part of calculus. They are used to calculate the areas, volumes, etc of arbitrary shapes for which formulas are not defined. Analytically they are just indefinite integrals with limits on top of them, but graphically they represent the area under the curve. The limits denote the boundaries between which the area should be calculated. These concepts hold a lot of importance in the field of electrical engineering, robotics, etc. For defining integrals, Riemann sums are used in which we calculate the area under any curve using infinitesimally small rectangles. Let’s look at this interpretation of definite integrals in detail. </p><div id="GFG_AD_gfg_mobile_336x280" style="margin: 5px 0;"></div> <img src="https://media.geeksforgeeks.org/wp-content/uploads/20210610022135/graph5.PNG" srcset="https://media.geeksforgeeks.org/wp-content/uploads/20210610022135/graph5.PNG, " sizes="100vw" width="479"> <p>Now this area can be divided into a number of rectangles, assuming the area is divided into ‘n’ rectangles of equal width. Some of these rectangles go above the actual value of the function and some go below, which means that the area is overestimated and underestimated in certain parts. The height of these rectangles is determined by the value of the function in the middle of the interval. </p> <img src="https://media.geeksforgeeks.org/wp-content/uploads/20210610022208/graph6.PNG" srcset="https://media.geeksforgeeks.org/wp-content/uploads/20210610022208/graph6.PNG, " sizes="100vw" width="476"> <p>In the definite integral notation, this area will be represented as, </p> <p style="text-align:center;"><img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-1c051911cea5af00e8c602c3af92b87c_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\int^{b}_{a}f(x)dx" title="Rendered by QuickLaTeX.com" height="36" width="110" style="vertical-align: -9px;"/></p> <p>This area can be approximated by dividing the area under the curve into n equally sized rectangles. So, the interval [a, b] is divided into n-subintervals defined by the points. </p> <p style="text-align:center;">a = x<sub>0</sub> < x<sub>1 </sub>< x<sub>2 </sub>< …. x<sub>n-2</sub>< x<sub>n-1</sub> < x<sub>n </sub>= b</p><div id="GFG_AD_Desktop_InContent_ATF_336x280" style="text-align:center; max-height: 280px;"></div><div id="GFG_AD_gfg_mobile_336x280_1" style="margin: 5px 0;"></div> <p>Then, the n intervals are, </p><div id="_GFG_ABP_Incontent_728x90" style="text-align:center;"></div> <p>[x<sub>0</sub>, x<sub>1</sub>], [x<sub>1</sub>, x<sub>2</sub>], …. [x<sub>n-1</sub>, x<sub>n</sub>]</p> <p>So, for the i<sup>th</sup> rectangle, the width will be, [x<sub>i-1</sub>, x<sub>i</sub>]. </p> <p>The area for i<sup>th</sup> rectangle A<sub>i</sub> = f(<img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-de8c65cb4ed20de87e70c4fcaf8a662d_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\frac{x_{i-1} + x_{i}}{2}" title="Rendered by QuickLaTeX.com" height="34" width="77" style="vertical-align: -9px;"/>)(x<sub>i —</sub> x<sub>i-1</sub>)</p> <p>So, the total area will be <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-7b88aeacf72e47f1cefc9ff9700df94a_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\sum^{n}_{i = 1}A_{i}" title="Rendered by QuickLaTeX.com" height="29" width="93" style="vertical-align: -8px;"/></p> <p>This sum is called <strong>Riemann sum</strong>.</p><div style="text-align:center;margin:20px 0; max-height: 280px;" id="GFG_AD_gfg_outstream_incontent"></div> <h3><strong>Riemann Sums in Sigma Notation</strong></h3> <p>Assume that the goal is to calculate the area under the graph of the function f(x) = x<sup>2</sup>, the area will be calculated between the limits x = 0.5 to x = 4.5.</p> <img src="https://media.geeksforgeeks.org/wp-content/uploads/20210610022731/graph7.PNG" srcset="https://media.geeksforgeeks.org/wp-content/uploads/20210610022731/graph7.PNG, " sizes="100vw" width="447"> <p>Divide the interval into four equal parts, the intervals will be [0.5, 1.5], [1.5, 2.5], [2.5, 3.5] and [3.5, 4.5]. </p> <img src="https://media.geeksforgeeks.org/wp-content/uploads/20210610022853/graph8.PNG" srcset="https://media.geeksforgeeks.org/wp-content/uploads/20210610022853/graph8.PNG, " sizes="100vw" width="495"> <p>The riemann sum then, can be written as follows, </p> <p>A(1) + A(2) + A(3) + A(4) = <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-27cb24abd12bb03213854fd984d6eabb_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\sum^{4}_{i=1}A_{i}" title="Rendered by QuickLaTeX.com" height="33" width="93" style="vertical-align: -8px;"/></p> <p>Let the heights of the interval be the values of the function at the end of the rectangle. This is called the <strong>right sum Riemann sum</strong>. Let x<sub>i </sub>denote the right endpoint of the i<sup>th</sup> rectangle. </p> <p>So, the formula for x<sub>i </sub>= 0.5 + i. Now, the value of the function at these points becomes, </p> <p>f(x<sub>i</sub>) = (0.5 + i)<sup>2</sup></p> <p>So, A(i) = (height)(width) </p> <p>            = (0.5 + i)<sup>2</sup></p> <p>The Riemann sum becomes, </p><div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-1"></div> <p> A(1) + A(2) + A(3) + A(4) = <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-27cb24abd12bb03213854fd984d6eabb_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\sum^{4}_{i=1}A_{i}" title="Rendered by QuickLaTeX.com" height="33" width="93" style="vertical-align: -8px;"/></p> <p>⇒ A(1) + A(2) + A(3) + A(4) = <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-9005461b87538d6eb7134fe5566b3f6f_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\sum^{4}_{i=1}(0.5 + i)^2" title="Rendered by QuickLaTeX.com" height="33" width="166" style="vertical-align: -8px;"/></p> <p>So, this way almost all the riemann sums can be represented in a sigma notation. </p> <p>To summarize the whole process, </p> <blockquote> <p><strong>Step 1:</strong> Find out the width of each interval. Let’s denote the width of interval with <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-88e72931a47ff327bb194069ff8e4d7f_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\Delta x" title="Rendered by QuickLaTeX.com" height="19" width="35" style="vertical-align: 0px;"/></p> <p><strong>Step 2:</strong> Let x<sub>i</sub> denote the right-endpoint of the rectangle x<sub>i </sub>= a + <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-301dd9ed3d98f394171743a832e791df_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\Delta" title="Rendered by QuickLaTeX.com" height="19" width="20" style="vertical-align: 0px;"/> x.i</p> <p><strong>Step 3: </strong>Define the area of each rectangle. </p> <p><strong>Step 4: </strong>Sum the areas </p> </blockquote><h3>Sample Problems</h3> <p><strong>Question 1: Consider a function f(x), its area is calculated from riemann sum from x = 4 to x = 10, the whole area is divided into 5 rectangles. Find the interval width. </strong></p> <p><strong>Solution: </strong></p><div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-2"></div> <blockquote> <p>The whole length is divided into 5 equal parts, </p> <p>x<sub>i = </sub>4 and x<sub>l</sub> = 10, </p> <p>Width of an interval is given by = <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-d6d3417fccd6fde222566cfb08850df8_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\frac{x_{l} -x_{i}}{n}" title="Rendered by QuickLaTeX.com" height="30" width="54" style="vertical-align: -9px;"/></p> <p>Where x<sub>i </sub>– initial point, and x<sub>l</sub> – last point and n= number of parts </p> <p>n = 5 </p> <p><img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-0b4a22a91a41b99f05472d91409df275_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\Delta x = \frac{x_l - x_i}{n} \\ = \frac{10 - 4}{5} \\ = \frac{6}{5}  \\ = 1.2" title="Rendered by QuickLaTeX.com" height="119" width="128" style="vertical-align: 0px;"/></p> </blockquote> <p><strong>Question 2: Consider a function f(x) = x, its area is calculated from riemann sum from x = 0 to x = 5, the whole area is divided into 5 rectangles. Find the riemann sum in sigma notation</strong></p> <p><strong>Solution: </strong></p> <blockquote> <p><strong>Step (i):</strong> Calculate the width</p> <p>The whole length is divided into 5 equal parts, </p><div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-3"></div> <p>x<sub>i </sub>= 0 and x<sub>l</sub> = 5, </p> <p>Width of an interval is given by = <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-d6d3417fccd6fde222566cfb08850df8_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\frac{x_{l} -x_{i}}{n}" title="Rendered by QuickLaTeX.com" height="30" width="54" style="vertical-align: -9px;"/></p> <p>Where x<sub>i </sub>= initial point, and x<sub>l</sub> – last point and n= number of parts </p> <p>n = 5 </p> <p><img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-34e9ccb69da0a72582a49d293b33e46f_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\Delta x = \frac{x_l - x_i}{n} \\ = \frac{5 - 0}{5} \\ = 1" title="Rendered by QuickLaTeX.com" height="86" width="128" style="vertical-align: 0px;"/></p> <p><strong>Step(ii)</strong>: </p> <p>a = 0, </p> <p>x<sub>i</sub> = 0 + <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-7f5671ea9bae10e61151cab63fb3f64f_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt=" \Delta x.i" title="Rendered by QuickLaTeX.com" height="19" width="51" style="vertical-align: 0px;"/></p> <p>⇒ x<sub>i</sub> = i</p> <p><strong>Step (iii) </strong></p><div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-4"></div> <p>A<sub>i</sub> = Height x Width </p> <p>    = f(x<sub>i</sub>) <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-bb74af691d2b8e4eb579e626f8dffd2c_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt=" \Delta x " title="Rendered by QuickLaTeX.com" height="19" width="35" style="vertical-align: 0px;"/></p> <p>    = (i)</p> <p>Total Area = A(1) + A(2) + A(3) + A(4) + A(5)</p> <p>                  = <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-9825540ffda8fe81b8aa7a07daca73b8_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\sum^{5}_{i=1}i" title="Rendered by QuickLaTeX.com" height="33" width="74" style="vertical-align: -8px;"/></p> </blockquote> <p><strong>Question 3: Consider a function f(x) = 5 – x, its area is calculated from riemann sum from x = 0 to x = 5, the whole area is divided into 5 rectangles. Find the riemann sum in sigma notation</strong></p> <p><strong>Solution: </strong></p> <blockquote> <p><strong>Step (i):</strong> Calculate the width</p> <p>The whole length is divided into 5 equal parts, </p> <p>x<sub>i </sub>= 0 and x<sub>l</sub> = 5, </p><div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-5"></div> <p>Width of an interval is given by = <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-d6d3417fccd6fde222566cfb08850df8_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\frac{x_{l} -x_{i}}{n}" title="Rendered by QuickLaTeX.com" height="30" width="54" style="vertical-align: -9px;"/></p> <p>Where x<sub>i </sub>= initial point, and x<sub>l</sub> – last point and n= number of parts </p> <p>n = 5 </p> <p><img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-34e9ccb69da0a72582a49d293b33e46f_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\Delta x = \frac{x_l - x_i}{n} \\ = \frac{5 - 0}{5} \\ = 1" title="Rendered by QuickLaTeX.com" height="86" width="128" style="vertical-align: 0px;"/></p> <p>Step(ii): </p> <p>a = 0, </p> <p>x<sub>i</sub> = 0 + <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-7f5671ea9bae10e61151cab63fb3f64f_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt=" \Delta x.i" title="Rendered by QuickLaTeX.com" height="19" width="51" style="vertical-align: 0px;"/></p> <p>⇒ x<sub>i</sub> = i</p> <p>Step (iii) </p> <p>A<sub>i</sub> = Height x Width </p><div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-6"></div> <p>    = f(x<sub>i</sub>) <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-bb74af691d2b8e4eb579e626f8dffd2c_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt=" \Delta x " title="Rendered by QuickLaTeX.com" height="19" width="35" style="vertical-align: 0px;"/></p> <p>    = (5 – i)(1)</p> <p>    = 5 – i</p> <p>Total Area = A(1) + A(2) + A(3) + A(4) + A(5)</p> <p>                  = <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-62d4d7d50fb6d76ad86ff1f73e8373a4_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\sum^{5}_{i=1}5 - i" title="Rendered by QuickLaTeX.com" height="33" width="119" style="vertical-align: -8px;"/></p> </blockquote> <p><strong>Question 4: Consider a function f(x) = √x, its area is calculated from riemann sum from x = 0 to x = 4, the whole area is divided into 4 rectangles. Find the riemann sum in sigma notation</strong></p> <p><strong>Solution:</strong></p> <blockquote> <p><strong>Step (i):</strong> Calculate the width</p> <p>The whole length is divided into 4 equal parts, </p> <p>x<sub>i </sub>= 0 and x<sub>l</sub> = 4, </p><div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-7"></div> <p>Width of an interval is given by = <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-d6d3417fccd6fde222566cfb08850df8_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\frac{x_{l} -x_{i}}{n}" title="Rendered by QuickLaTeX.com" height="30" width="54" style="vertical-align: -9px;"/></p> <p>Where x<sub>i </sub>= initial point, and x<sub>l</sub> – last point and n= number of parts </p> <p>n = 4 </p> <p><img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-59aed0b0af67c54006fda74cd4cb2a8c_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\Delta x = \frac{x_l - x_i}{n} \\ = \frac{4 - 0}{4} \\ = 1" title="Rendered by QuickLaTeX.com" height="86" width="128" style="vertical-align: 0px;"/></p> <p><strong>Step(ii): </strong></p> <p>a = 0, </p> <p>x<sub>i</sub> = 0 + <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-c1c92c727d80d6d66409e13a078aee8c_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\Delta x.i" title="Rendered by QuickLaTeX.com" height="19" width="51" style="vertical-align: 0px;"/></p> <p>⇒ x<sub>i</sub> = i</p> <p>Step (iii) </p> <p>A<sub>i</sub> = Height × Width </p><div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-8"></div> <p>    = f(x<sub>i</sub>) <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-88e72931a47ff327bb194069ff8e4d7f_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\Delta x" title="Rendered by QuickLaTeX.com" height="19" width="35" style="vertical-align: 0px;"/></p> <p>    = (√i)(1)</p> <p>    = √i</p> <p>Total Area = A(1) + A(2) + A(3) + A(4) </p> <p>                  = <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-278011feca03925b42dcd4df9aad71dc_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\sum^{4}_{i=1}\sqrt{i}" title="Rendered by QuickLaTeX.com" height="33" width="98" style="vertical-align: -8px;"/></p> </blockquote> <p><strong>Question 5: Consider a function f(x) = e<sup>x</sup> + 1, its area is calculated from riemann sum from x = 0 to x = 4, the whole area is divided into 4 rectangles. Find the riemann sum in sigma notation</strong></p> <p><strong>Solution:</strong></p> <blockquote> <p><strong>Step (i): </strong>Calculate the width</p> <p>The whole length is divided into 4 equal parts, </p> <p>x<sub>i </sub>= 0 and x<sub>l</sub> = 4, </p> <p>Width of an interval is given by = <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-d6d3417fccd6fde222566cfb08850df8_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\frac{x_{l} -x_{i}}{n}" title="Rendered by QuickLaTeX.com" height="30" width="54" style="vertical-align: -9px;"/></p> <p>Where x<sub>i </sub>= initial point, and x<sub>l</sub> – last point and n= number of parts </p> <p>n = 4 </p> <p><img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-59aed0b0af67c54006fda74cd4cb2a8c_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\Delta x = \frac{x_l - x_i}{n} \\ = \frac{4 - 0}{4} \\ = 1" title="Rendered by QuickLaTeX.com" height="86" width="128" style="vertical-align: 0px;"/></p> <p>Step(ii): </p> <p>a = 0, </p> <p>x<sub>i</sub> = 0 + <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-c1c92c727d80d6d66409e13a078aee8c_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\Delta x.i" title="Rendered by QuickLaTeX.com" height="19" width="51" style="vertical-align: 0px;"/></p> <p>⇒ x<sub>i</sub> = i</p> <p>Step (iii) </p> <p>A<sub>i</sub> = Height x Width </p><div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-9"></div> <p>    = f(x<sub>i</sub>) <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-bb74af691d2b8e4eb579e626f8dffd2c_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt=" \Delta x " title="Rendered by QuickLaTeX.com" height="19" width="35" style="vertical-align: 0px;"/></p> <p>    = (e<sup>i</sup> + 1)(1)</p> <p>    = e<sup>i</sup> + 1</p> <p>Total Area = A(1) + A(2) + A(3) + A(4) </p> <p>                  = <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-16c4526b4cb062d0e42496c8782821ae_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\sum^{4}_{i=1}(e^i + 1)" title="Rendered by QuickLaTeX.com" height="33" width="145" style="vertical-align: -8px;"/></p> </blockquote> <p><strong>Question 6: Consider a function f(x) = x<sup>2</sup> + x, its area is calculated from riemann sum from x = 0 to x = 6, the whole area is divided into 3 rectangles. Find the riemann sum in sigma notation</strong></p> <p><strong>Solution:</strong></p> <blockquote> <p><strong>Step (i):</strong> Calculate the width</p> <p>The whole length is divided into 4 equal parts, </p> <p>x<sub>i </sub>= 0 and x<sub>l</sub> = 6, </p> <p>Width of an interval is given by = <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-d6d3417fccd6fde222566cfb08850df8_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\frac{x_{l} -x_{i}}{n}" title="Rendered by QuickLaTeX.com" height="30" width="54" style="vertical-align: -9px;"/></p> <p>Where x<sub>i </sub>= initial point, and x<sub>l</sub> – last point and n= number of parts </p> <p>n = 3 </p> <p><img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-743a52ac6fce8add1b155e213a2bdc75_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\Delta x = \frac{x_l - x_i}{n} \\ = \frac{6 - 0}{3} \\ = 2" title="Rendered by QuickLaTeX.com" height="86" width="128" style="vertical-align: 0px;"/></p> <p><strong>Step(ii): </strong></p> <p>a = 0, </p> <p>x<sub>i</sub> = 0 + <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-c1c92c727d80d6d66409e13a078aee8c_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\Delta x.i" title="Rendered by QuickLaTeX.com" height="19" width="51" style="vertical-align: 0px;"/></p> <p>⇒ x<sub>i</sub> = 2i</p> <p>Step (iii) </p> <p>A<sub>i</sub> = Height x Width </p> <p>    = f(x<sub>i</sub>) <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-bb74af691d2b8e4eb579e626f8dffd2c_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt=" \Delta x " title="Rendered by QuickLaTeX.com" height="19" width="35" style="vertical-align: 0px;"/></p> <p>    = (i<sup>2</sup> + i)(2)</p> <p>Total Area = A(1) + A(2) + A(3) + A(4) + A(5) + A(6)</p> <p>                  = <img src="https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-b33ec4fff9d0f8373959fe924d21f4da_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\sum^{4}_{i=1}(i^2 + i)" title="Rendered by QuickLaTeX.com" height="33" width="141" style="vertical-align: -8px;"/></p> </blockquote><br/><div id="AP_G4GR_6"></div> <div class='article_bottom_text'></div><br> <div id="video-tab-content" class="video-tab-content"> <div style="text-align: center; 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Evaluating composite f</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/invertible-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Invertible Functions </div> <div class="gfg-similar-read-item-subheading">As the name suggests Invertible means "inverse", and Invertible function means the inverse of the function. Invertible functions, in the most general sense, are functions that "reverse" each other. For example, if f takes a to b, then the inverse, f-1, must take b to a. Table of Content Invertible F</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/composition-of-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Composition of Functions </div> <div class="gfg-similar-read-item-subheading">Composition of Function is the process or operation which combines two or more functions together into a single function. We define functions as the set of operations that operate on a set of values and give the desired output. Suppose f is a function say a function which draws the sketch of the fru</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/inverse-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Inverse Functions | Definition, Condition for Inverse and Examples </div> <div class="gfg-similar-read-item-subheading">Inverse Functions are an important concept in mathematics. An inverse function basically reverses the effect of the original function. If you apply a function to a number and then apply its inverse, you get back the original number. For example, if a function turns 2 into 5, the inverse function wil</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/verifying-inverse-functions-by-composition/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Verifying Inverse Functions by Composition </div> <div class="gfg-similar-read-item-subheading">A function can be seen as a mathematical formula or a machine that throws output when an input is given. The output is usually some processed version of the input. Function's inverses can be seen as the operations which give us the input back on giving them the output. In other words, inverse functi</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 2: Inverse Trigonometric Functions<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/inverse-trigonometric-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Inverse Trigonometric Functions | Definition, Formula, Types and Examples </div> <div class="gfg-similar-read-item-subheading">Inverse trigonometric functions are the inverse functions of basic trigonometric functions. In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. The inverse trigonometric functions are the inverse functions of basic trigonometric function</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/graphs-of-inverse-trigonometric-functions-trigonometry-class-12-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Graphs of Inverse Trigonometric Functions - Trigonometry | Class 12 Maths </div> <div class="gfg-similar-read-item-subheading">Inverse trigonometric functions are the inverse functions of the trigonometric ratios i.e. sin, cos, tan, cot, sec, cosec. These functions are widely used in fields like physics, mathematics, engineering and other research fields. There are two popular notations used for inverse trigonometric functi</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/properties-of-inverse-trigonometric-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Properties of Inverse Trigonometric Functions </div> <div class="gfg-similar-read-item-subheading">Inverse trigonometric functions, also known as arc functions, are used to find the angle that corresponds to a given trigonometric value. These functions are the inverses of the six main trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. In this article, we will explore</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/inverse-trigonometric-identities/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Inverse Trigonometric Identities </div> <div class="gfg-similar-read-item-subheading">Inverse Trigonometric Identities: In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. The inverse trigonometric functions are the inverse functions of basic trigonometric functions, i.e., sine, cosine, tangent, cosecant, secant, and cota</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 3: Matrices<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/types-of-matrices/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Types of Matrices </div> <div class="gfg-similar-read-item-subheading">Types of Matrices classify matrices into different categories based on the number of rows and columns present in them, the position of the elements, and also the special properties exhibited by the Matrix. A matrix is a rectangular array of numbers in which elements are arranged in rows and columns.</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/matrix-operations/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Matrix Operations </div> <div class="gfg-similar-read-item-subheading">Matrix Operations are the operations that are operated on the matrix. Matrix Operation includes operations such as Addition of Matrix, Subtraction of Matrix, Multiplication of Matrix, etc, and others. These operations are very b useful for solving various problems of matrices and help us to find the</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/matrix-addition/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Matrix Addition </div> <div class="gfg-similar-read-item-subheading">Matrix Addition as the name suggests in the article, explores the addition of matrices and it is one of the fundamental operations in the field of Linear Algebra. Matrix Addition is very useful in various fields such as data analysis, computer graphics, image processing, cryptography, operations res</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/matrix-multiplication/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Matrix Multiplication </div> <div class="gfg-similar-read-item-subheading">Matrix multiplication is a fundamental operation in mathematics that involves multiplying two or more matrices by following specific rules. Understanding how to multiply matrices is crucial for solving various mathematical problems. In this article, we will learn how to do matrix multiplication, rul</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">14 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/transpose-of-a-matrix/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Transpose of a Matrix </div> <div class="gfg-similar-read-item-subheading">Transpose of a matrix is a very common method used for matrix transformation in linear algebra. Transpose of a matrix is obtained by interchanging the rows and columns of the given matrix or vice versa. Transpose of a matrix can be utilized to obtain the adjoint and inverse of the matrices. Before l</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/what-is-symmetric-matrix-and-skew-symmetric-matrix/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Symmetric and Skew Symmetric Matrices </div> <div class="gfg-similar-read-item-subheading">Symmetric and Skew Symmetric Matrices are the types of square matrices based on the relation between a matrix and its transpose. These matrices are one of the most used matrices out of all the matrices out there. Symmetric matrices have use cases in optimization, physics, and statistics, whereas ske</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/elementary-operations-on-matrices/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Elementary Operations on Matrices </div> <div class="gfg-similar-read-item-subheading">Elementary Operations on Matrices are the operations performed on the rows and columns of the matrix that do not change the value of the matrix. A matrix is a way of representing numbers in the form of an array, i.e. the numbers are arranged in the form of rows and columns. In a matrix, the rows and</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/inverse-of-a-matrix-by-elementary-operations-matrices-class-12-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Inverse of a Matrix by Elementary Operations - Matrices | Class 12 Maths </div> <div class="gfg-similar-read-item-subheading">The Gaussian Elimination method is also known as the row reduction method and it is an algorithm that is used to solve a system of linear equations. It is usually understood as a sequence of operations performed on the corresponding matrix of coefficients. Inverse of a MatrixThe inverse of a matrix</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/invertible-matrix/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Invertible Matrix </div> <div class="gfg-similar-read-item-subheading">Invertible matrices are defined as the matrix whose inverse exists. We define a matrix as the arrangement of data in rows and columns, if any matrix has m rows and n columns then the order of the matrix is m × n where m and n represent the number of rows and columns respectively. We define invertib</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 4: Determinants<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/what-is-determinant-of-a-matrix/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Determinant of Matrix with Solved Examples </div> <div class="gfg-similar-read-item-subheading">Determinant of a matrix is a special number that can be calculated for a square matrix (a matrix with the same number of rows and columns). It is a function that gives the unique output (real number) for every input value of the square matrix. The determinant of the matrix is considered the scaling</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/properties-of-determinants-class-12-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Properties of Determinants </div> <div class="gfg-similar-read-item-subheading">Properties of Determinants are the properties that are required to solve various problems in Matrices. There are various properties of the determinant that are based on the elements, rows, and columns of the determinant. These properties help us to easily find the value of the determinant. Suppose w</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/area-of-triangle-using-determinants/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Area of a Triangle using Determinants </div> <div class="gfg-similar-read-item-subheading">Area of a Triangle using Determinants is one of the multiple methods available to find the area of a triangle and this method is often overlooked for its complexity. But the Area of a Triangle using Determinants comes very handy in coordinate geometry when all three vertices of a triangle are given</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/minors-and-cofactors/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Minors and Cofactors </div> <div class="gfg-similar-read-item-subheading">Minors and Cofactors are important to calculate the adjoint and inverse of a matrix. As the name suggests, a Minor is a smaller part of the larger matrix obtained for a particular element of the matrix by deleting the terms of the row and column to which the element belongs. A cofactor is (-1)i+j ti</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/adjoint-of-matrix/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Adjoint of a Matrix </div> <div class="gfg-similar-read-item-subheading">Knowledge of matrices is necessary for various branches of mathematics. Matrices are one of the most powerful tools in mathematics. From matrices there come Determinants, Now we see one of the properties of the Determinant in this article. In this article, we see how to find the Adjoint of a Matrix.</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/applications-of-matrices-and-determinants/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Applications of Matrices and Determinants </div> <div class="gfg-similar-read-item-subheading">Applications of Matrices and Determinants: One application of matrices and determinants is that they can be used to solve linear equations in two or three variables. Matrices and determinants are also used to check the consistency of any system, whether they are consistent or not. This is the most u</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 5: Continuity and Differentiability<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/continuity-and-discontinuity-in-calculus-class-12-cbse/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Continuity and Discontinuity in Calculus </div> <div class="gfg-similar-read-item-subheading">Continuity and Discontinuity: Continuity and discontinuity are fundamental concepts in calculus and mathematical analysis, describing the behavior of functions. A function is continuous at a point if you can draw the graph of the function at that point without lifting your pen from the paper. Contin</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/differentiability-of-a-function-class-12-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Differentiability of a Function | Class 12 Maths </div> <div class="gfg-similar-read-item-subheading">Continuity or continuous which means, "a function is continuous at its domain if its graph is a curve without breaks or jumps". A function is continuous at a point in its domain if its graph does not have breaks or jumps in the immediate neighborhood of the point. Continuity at a Point: A function f</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/derivatives-of-inverse-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Derivatives of Inverse Functions </div> <div class="gfg-similar-read-item-subheading">In mathematics, a function(e.g. f), is said to be an inverse of another(e.g. g), if given the output of g returns the input value given to f. Additionally, this must hold true for every element in the domain co-domain(range) of g. E.g. assuming x and y are constants if g(x) = y and f(y) = x then the</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/derivatives-of-implicit-functions-continuity-and-differentiability-class-12-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Derivatives of Implicit Functions - Continuity and Differentiability | Class 12 Maths </div> <div class="gfg-similar-read-item-subheading">Implicit functions are functions where a specific variable cannot be expressed as a function of the other variable. A function that depends on more than one variable. Implicit Differentiation helps us compute the derivative of y with respect to x without solving the given equation for y, this can be</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/derivatives-of-composite-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Derivatives of Composite Functions </div> <div class="gfg-similar-read-item-subheading">Derivatives are an essential part of calculus. They help us in calculating the rate of change, maxima, and minima for the functions. Derivatives by definition are given by using limits, which is called the first form of the derivative. We already know how to calculate the derivatives for standard fu</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/derivatives-of-inverse-trigonometric-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Derivatives of Inverse Trigonometric Functions </div> <div class="gfg-similar-read-item-subheading">Derivatives of Inverse Trigonometric Functions: Every mathematical function, from the simplest to the most complex, has an inverse. 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We know that exponential functions exist in two forms, ax where a is a real number r and is greater than 0 and the other form is ex where e is Euler's Number and the value of e is 2.718 . . .</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/logarithmic-differentiation-continuity-and-differentiability/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Logarithmic Differentiation - Continuity and Differentiability </div> <div class="gfg-similar-read-item-subheading">The word continuity means something which is continuous in nature. The flow of water is continuous, time in real life is continuous, and many more instances show the continuity in real life. In mathematics, the Continuous function is the one which when drawn on a graph does not show any breaks and i</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/proofs-for-the-derivatives-of-e%cb%a3-and-lnx-advanced-differentiation/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Proofs for the derivatives of eˣ and ln(x) - Advanced differentiation </div> <div class="gfg-similar-read-item-subheading">In this article, we are going to cover the proofs of the derivative of the functions ln(x) and ex. 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They are used to solve various types of problems in Mathematics. Mean Value Theorem is also called Lagrenges's Mean Value Theorem. Rolle’s Theorem is a subcase of the mean value t</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/derivative-of-functions-in-parametric-form/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Derivative of Functions in Parametric Forms </div> <div class="gfg-similar-read-item-subheading">Parametric Differentiation refers to the differentiation of a function in which the dependent and independent variables are equated to a third variable. Derivatives of the functions express the rate of change in the functions. We know how to calculate the derivatives for standard functions. Chain ru</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/second-order-derivatives/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Second Order Derivatives: Rules , Formula and Examples (Class 12 Maths) </div> <div class="gfg-similar-read-item-subheading">The Second Order Derivative is defined as the derivative of the first derivative of the given function. 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Mean Value Theorem is abbreviated as MVT. This theorem was first proposed by an Indian Mathem</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/algebra-of-continuous-functions-continuity-and-differentiability-class-12-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Algebra of Continuous Functions - Continuity and Differentiability | Class 12 Maths </div> <div class="gfg-similar-read-item-subheading">Algebra of Continuous Functions deals with the utilization of continuous functions in equations involving the varied binary operations you've got studied so. We'll also mention a composition rule that may not be familiar to you but is extremely important for future applications. Since the continuity</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 6: Applications of Derivatives<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/critical-points/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Critical Points </div> <div class="gfg-similar-read-item-subheading">As the complexity of the functions increase, we see more and more complex behavior from their graphs, and it becomes harder to graph. There have lots of peaks and valleys in their graphs. It becomes essential to find out the position of these valleys and peaks, the peaks are called maxima and the va</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/derivatives-as-rate-of-change/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Derivatives as Rate of Change </div> <div class="gfg-similar-read-item-subheading">Derivatives are a mathematical tool used to analyze how quantities change. We can calculate derivatives for various, quotient, and chain rulesfunctions, including trigonometric, exponential, polynomial, and implicit functions. There are two main methods for calculating derivatives: using limits or a</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/increasing-and-decreasing-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Increasing and Decreasing Functions </div> <div class="gfg-similar-read-item-subheading">If you're studying calculus, then you're probably familiar with the concepts of increasing and decreasing functions. These terms refer to the behavior of a function as its input values change. Specifically, an increasing function is one that becomes larger as its input values increase, while a decre</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/increasing-and-decreasing-intervals/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Increasing and Decreasing Intervals </div> <div class="gfg-similar-read-item-subheading">Increasing and decreasing intervals are the intervals of real numbers in which real-valued functions are increasing and decreasing respectively. Derivatives are a way of measuring the rate of change of a variable. When it comes to functions and calculus, derivatives give us a lot of information abou</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/tangents-and-normals/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Tangents and Normals </div> <div class="gfg-similar-read-item-subheading">Tangent and Normals are the lines that are used to define various properties of the curves. We define tangent as the line which touches the circle only at one point and normal is the line that is perpendicular to the tangent at the point of tangency. Any tangent of the curve passing through the poin</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/equation-of-tangents-and-normals/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Equation of Tangents and Normals </div> <div class="gfg-similar-read-item-subheading">Derivatives are used to find rate of change of a function with respect to variables. To find rate of change of function with respect to a variable differentiating it with respect to that variable is required. Rate of change of function y = f(x) with respect to x is defined by dy/dx or f'(x). For exa</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/relative-minima-and-maxima/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Relative Minima and Maxima </div> <div class="gfg-similar-read-item-subheading">Relative maxima and minima are the points defined in any function such that at these points the value of the function is either maximum or minimum in their neighborhood. Relative maxima and minima depend on their neighborhood point and are calculated accordingly. We find the relative maxima and mini</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/absolute-minima-and-maxima/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Absolute Minima and Maxima </div> <div class="gfg-similar-read-item-subheading">Absolute Maxima and Minima are the maximum and minimum values of the function defined on a fixed interval. A function in general can have high values or low values as we move along the function. The maximum value of the function in any interval is called the maxima and the minimum value of the funct</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/concave-function/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Concave Function </div> <div class="gfg-similar-read-item-subheading">Graphs of the functions give us a lot of information about the nature of the function, the trends, and the critical points like maxima and minima of the function. Derivatives allow us to mathematically analyze these functions and their sign can give us information about the maximum and minimum of th</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/inflection-point/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Inflection Point </div> <div class="gfg-similar-read-item-subheading">Inflection Point describes a point where the curvature of a curve changes direction. It represents the transition from a concave to a convex shape or vice versa. Let's learn about Inflection Points in detail, including Concavity of Function and solved examples. Table of Content Inflection Point Defi</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/curve-sketching/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Curve Sketching </div> <div class="gfg-similar-read-item-subheading">Curve Sketching as its name suggests helps us sketch the approximate graph of any given function which can further help us visualize the shape and behavior of a function graphically. Curve sketching isn't any sure-shot algorithm that after application spits out the graph of any desired function but</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/approximations-maxima-and-minima-application-of-derivatives-class-12-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Approximations & Maxima and Minima - Application of Derivatives | Class 12 Maths </div> <div class="gfg-similar-read-item-subheading">An approximation is similar but not exactly equal to something else. Approximation occurs when an exact numerical number is unknown or difficult to obtain. In Mathematics, we use differentiation to find the approximate values of certain quantities. Let f be a given function and let y = f(x). Let ∆x</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/higher-order-derivatives/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Higher Order Derivatives </div> <div class="gfg-similar-read-item-subheading">Higher Order Derivatives are the second, third, or further derivative of the function, i.e. differentiating a function multiple times results in a higher order derivative. Suppose we have a function f(x) then its differentiation is f'(x) which is a first-order derivative. Then its differentiation ag</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown" style="background-color:var(--leftbar-background-color); border-bottom:1px solid #CCC;"><h2 class="dropdown-title active">Chapter 7: Integrals<span class="gfg-icon gfg-icon_arrow-down-thin upside"></span></h2><ul><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/integrals/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Integrals </div> <div class="gfg-similar-read-item-subheading">Integrals: An integral in mathematics is a continuous analog of a sum that is used to determine areas, volumes, and their generalizations. Performing integration is the process of computing an integral and is one of the two basic concepts of calculus. Integral in Calculus is the branch of Mathematic</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/integration-by-substitution/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Integration by Substitution Method </div> <div class="gfg-similar-read-item-subheading">Integration by substitution is one of the important methods for finding the integration of the function where direct integration can not be easily found. This method is very useful in finding the integration of complex functions. We use integration by substitution to reduce the given function into t</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/integration-by-partial-fractions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Integration by Partial Fractions </div> <div class="gfg-similar-read-item-subheading">Integration by Partial Fractions is one of the methods of integration, which is used to find the integral of the rational functions. In Partial Fraction decomposition, an improper-looking rational function is decomposed into the sum of various proper rational functions. If f(x) and g(x) are polynomi</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/integration-by-parts/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Integration by Parts </div> <div class="gfg-similar-read-item-subheading">Integration by Parts: Integration by parts is a technique used in calculus to find the integral of the product of two functions. It's essentially a reversal of the product rule for differentiation. Integrating a function is not always easy sometimes we have to integrate a function that is the multip</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">14 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/integration-of-trigonometric-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Integration of Trigonometric Functions </div> <div class="gfg-similar-read-item-subheading">Integration is the process of summing up small values of a function in the region of limits. It is just the opposite to differentiation. Integration is also known as anti-derivative. We have explained the Integration of Trigonometric Functions in this article below. Below is an example of the Integr</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/functions-defined-by-integrals/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Functions Defined by Integrals </div> <div class="gfg-similar-read-item-subheading">While thinking about functions, we always imagine that a function is a mathematical machine that gives us an output for any input we give. It is usually thought of in terms of mathematical expressions like squares, exponential and trigonometric function, etc. It is also possible to define the functi</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/definite-integral/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Definite Integral | Definition, Formula & How to Calculate </div> <div class="gfg-similar-read-item-subheading">A definite integral is an integral that calculates a fixed value for the area under a curve between two specified limits. The resulting value represents the sum of all infinitesimal quantities within these boundaries. i.e. if we integrate any function within a fixed interval it is called a Definite</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/computing-definite-integrals/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Computing Definite Integrals </div> <div class="gfg-similar-read-item-subheading">Integrals are a very important part of the calculus. They allow us to calculate the anti-derivatives, that is given a function's derivative, integrals give the function as output. Other important applications of integrals include calculating the area under the curve, the volume enclosed by a surface</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/fundamental-theorem-of-calculus/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Fundamental Theorem of Calculus | Part 1, Part 2 </div> <div class="gfg-similar-read-item-subheading">Fundamental Theorem of Calculus is the basic theorem that is widely used for defining a relation between integrating a function of differentiating a function. The fundamental theorem of calculus is widely useful for solving various differential and integral problems and making the solution easy for</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/finding-derivative-with-fundamental-theorem-of-calculus/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Finding Derivative with Fundamental Theorem of Calculus </div> <div class="gfg-similar-read-item-subheading">Integrals are the reverse process of differentiation. They are also called anti-derivatives and are used to find the areas and volumes of the arbitrary shapes for which there are no formulas available to us. Indefinite integrals simply calculate the anti-derivative of the function, while the definit</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/evaluating-definite-integrals/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Evaluating Definite Integrals </div> <div class="gfg-similar-read-item-subheading">Integration, as the name suggests is used to integrate something. In mathematics, integration is the method used to integrate functions. The other word for integration can be summation as it is used, to sum up, the entire function or in a graphical way, used to find the area under the curve function</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/properties-of-definite-integrals/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Properties of Definite Integrals </div> <div class="gfg-similar-read-item-subheading">Properties of Definite Integrals: An integral that has a limit is known as a definite integral. It has an upper limit and a lower limit. It is represented as [Tex]\int_{a}^{b}[/Tex]f(x) = F(b) − F(a) There are many properties regarding definite integral. We will discuss each property one by one with</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/definite-integrals-of-piecewise-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Definite Integrals of Piecewise Functions </div> <div class="gfg-similar-read-item-subheading">Imagine a graph with a function drawn on it, it can be a straight line or a curve or anything as long as it is a function. Now, this is just one function on the graph, can 2 functions simultaneously occur on the graph? Imagine two functions simultaneously occurring on the graph, say, a straight line</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/improper-integrals/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Improper Integrals </div> <div class="gfg-similar-read-item-subheading">Geometrically speaking, integrals are the way to compute the area or volume under curves. These methods allow mathematicians to compute the area under arbitrarily complex curves. These types of integrals are called definite integrals. Definite integrals are built upon the idea of indefinite integral</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/riemann-sum/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Riemann Sums </div> <div class="gfg-similar-read-item-subheading">Riemann Sum is a certain kind of approximation of an integral by a finite sum. A Riemann sum is the sum of rectangles or trapezoids that approximate vertical slices of the area in question. German mathematician Bernhard Riemann developed the concept of Riemann Sums. In this article, we will look int</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="currentpage similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/riemann-sums-in-summation-notation/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Riemann Sums in Summation Notation </div> <div class="gfg-similar-read-item-subheading">Riemann sums allow us to calculate the area under the curve for any arbitrary function. These formulations help us define the definite integral. The basic idea behind these sums is to divide the area that is supposed to be calculated into small rectangles and calculate the sum of their areas. These</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/trapezoidal-rule/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Trapezoidal Rule </div> <div class="gfg-similar-read-item-subheading">The trapezoidal rule is one of the fundamental rules of integration which is used to define the basic definition of integration. It is a widely used rule and the Trapezoidal rule is named so because it gives the area under the curve by dividing the curve into small trapezoids instead of rectangles.</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/definite-integral-as-the-limit-of-a-riemann-sum/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Definite Integral as the Limit of a Riemann Sum </div> <div class="gfg-similar-read-item-subheading">Definite integrals are an important part of calculus. They are used to calculate the areas, volumes, etc of arbitrary shapes for which formulas are not defined. Analytically they are just indefinite integrals with limits on top of them, but graphically they represent the area under the curve. The li</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/antiderivatives/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Antiderivatives </div> <div class="gfg-similar-read-item-subheading">Antiderivatives: The Antiderivative of a function is the inverse of the derivative of the function. Antiderivative is also called the Integral of a function. Suppose the derivative of a function d/dx[f(x)] is F(x) + C then the antiderivative of [F(x) + C] dx of the F(x) + C is f(x). An example expla</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/indefinite-integrals/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Integration Formulas </div> <div class="gfg-similar-read-item-subheading">Indefinite Integrals: The derivatives have been really useful in almost every aspect of life. They allow for finding the rate of change of a function. Sometimes there are situations where the derivative of a function is available, and the goal is to calculate the actual function whose derivative is</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/particular-solutions-to-differential-equations/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Particular Solutions to Differential Equations </div> <div class="gfg-similar-read-item-subheading">Indefinite integrals are the reverse of the differentiation process. Given a function f(x) and it's derivative f'(x), they help us in calculating the function f(x) from f'(x). These are used almost everywhere in calculus and are thus called the backbone of the field of calculus. Geometrically speaki</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/integration-by-u-substitution/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Integration by U-substitution </div> <div class="gfg-similar-read-item-subheading">Finding integrals is basically a reverse differentiation process. That is why integrals are also called anti-derivatives. Often the functions are straightforward and standard functions that can be integrated easily. It is easier to solve the combination of these functions using the properties of ind</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/reverse-chain-rule/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Reverse Chain Rule </div> <div class="gfg-similar-read-item-subheading">Integrals are an important part of the theory of calculus. They are very useful in calculating the areas and volumes for arbitrarily complex functions, which otherwise are very hard to compute and are often bad approximations of the area or the volume enclosed by the function. Integrals are the reve</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/partial-fraction-expansion/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Partial Fraction Expansion </div> <div class="gfg-similar-read-item-subheading">If f(x) is a function that is required to be integrated, f(x) is called the Integrand, and the integration of the function without any limits or boundaries is known as the Indefinite Integration. Indefinite integration has its own formulae to make the process of integration easier. However, sometime</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/trigonometric-substitution/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Trigonometric Substitution: Method, Formula and Solved Examples </div> <div class="gfg-similar-read-item-subheading">Trigonometric Substitution is one of the substitution methods of integration where a function or expression in the given integral is substituted with trigonometric functions such as sin, cos, tan, etc. Integration by substitution is the easiest substitution method. It is used when we make a substitu</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 8: Applications of Integrals<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/area-under-simple-curves/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Area under Simple Curves </div> <div class="gfg-similar-read-item-subheading">We know how to calculate the areas of some standard curves like rectangles, squares, trapezium, etc. There are formulas for areas of each of these figures, but in real life, these figures are not always perfect. Sometimes it may happen that we have a figure that looks like a square but is not actual</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/area-between-two-curves/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Area Between Two Curves: Formula, Definition and Examples </div> <div class="gfg-similar-read-item-subheading">Area Between Two Curves in Calculus is one of the applications of Integration. It helps us calculate the area bounded between two or more curves using the integration. As we know Integration in calculus is defined as the continuous summation of very small units. The topic "Area Between Two Curves" h</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/area-between-polar-curves/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Area between Polar Curves </div> <div class="gfg-similar-read-item-subheading">Coordinate systems allow the mathematical formulation of the position and behavior of a body in space. These systems are used almost everywhere in real life. Usually, the rectangular Cartesian coordinate system is seen, but there is another type of coordinate system which is useful for certain kinds</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/area-as-definite-integral/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Area as Definite Integral </div> <div class="gfg-similar-read-item-subheading">Integrals are an integral part of calculus. They represent summation, for functions which are not as straightforward as standard functions, integrals help us to calculate the sum and their areas and give us the flexibility to work with any type of function we want to work with. The areas for the sta</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 9: Differential Equations<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/differential-equations/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Differential Equations </div> <div class="gfg-similar-read-item-subheading">A differential equation is a mathematical equation that relates a function with its derivatives. Differential Equations come into play in a variety of applications such as Physics, Chemistry, Biology, Economics, etc. Differential equations allow us to predict the future behavior of systems by captur</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/particular-solutions-to-differential-equations/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Particular Solutions to Differential Equations </div> <div class="gfg-similar-read-item-subheading">Indefinite integrals are the reverse of the differentiation process. Given a function f(x) and it's derivative f'(x), they help us in calculating the function f(x) from f'(x). These are used almost everywhere in calculus and are thus called the backbone of the field of calculus. Geometrically speaki</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/homogeneous-differential-equations/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Homogeneous Differential Equations </div> <div class="gfg-similar-read-item-subheading">Homogeneous Differential Equations are differential equations with homogenous functions. They are equations containing a differentiation operator, a function, and a set of variables. The general form of the homogeneous differential equation is f(x, y).dy + g(x, y).dx = 0, where f(x, y) and h(x, y) i</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/separable-differential-equations/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Separable Differential Equations </div> <div class="gfg-similar-read-item-subheading">Separable differential equations are a special type of ordinary differential equation (ODE) that can be solved by separating the variables and integrating each side separately. Any differential equation that can be written in form of y' = f(x).g(y), is called a separable differential equation. Basic</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/exact-equations-and-integrating-factors/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Exact Equations and Integrating Factors </div> <div class="gfg-similar-read-item-subheading">Differential Equations are used to describe a lot of physical phenomena. They help us to observe something happening in real life and put it in a mathematical form. At this level, we are mostly concerned with linear and first-order differential equations. A differential equation in “y” is linear if</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/implicit-differentiation/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Implicit Differentiation </div> <div class="gfg-similar-read-item-subheading">Implicit Differentiation is a useful tool in the arsenal of tools to tackle problems in calculus and beyond which helps us differentiate the function without converting it into the explicit function of the independent variable. Suppose we don't know the method of implicit differentiation. In that ca</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/implicit-differentiation-advanced-examples/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Implicit differentiation - Advanced Examples </div> <div class="gfg-similar-read-item-subheading">In the previous article, we have discussed the introduction part and some basic examples of Implicit differentiation. So in this article, we will discuss some advanced examples of implicit differentiation. Table of Content Implicit DifferentiationMethod to solveImplicit differentiation Formula Solve</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/advanced-differentiation/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Advanced Differentiation </div> <div class="gfg-similar-read-item-subheading">Derivatives are used to measure the rate of change of any quantity. This process is called differentiation. It can be considered as a building block of the theory of calculus. Geometrically speaking, the derivative of any function at a particular point gives the slope of the tangent at that point of</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/disguised-derivatives-advanced-differentiation-class-12-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Disguised Derivatives - Advanced differentiation | Class 12 Maths </div> <div class="gfg-similar-read-item-subheading">The dictionary meaning of “disguise” is “unrecognizable”. Disguised derivative means “unrecognized derivative”. 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Sometimes finding the differentiation of the function is very tough but differentiating the logarithm of the same function is very easy, then in such cases, the logarithmic differenti</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 10: Vector Algebra<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/vector-algebra/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Vector Algebra </div> <div class="gfg-similar-read-item-subheading">Vectors algebra is the branch of algebra that involves operations on vectors. Vectors are quantities that have both magnitude and direction so normal operations are not performed on the vectors. We can add, subtract, and multiply vector quantities using special vector algebra rules. Vectors can be e</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">14 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/dot-and-cross-products-on-vectors/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Dot and Cross Products on Vectors </div> <div class="gfg-similar-read-item-subheading">A quantity that is characterized not only by magnitude but also by its direction, is called a vector. Velocity, force, acceleration, momentum, etc. are vectors. Vectors can be multiplied in two ways: Scalar product or Dot productVector Product or Cross productTable of Content Scalar Product/Dot Pr</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/how-to-find-the-angle-between-two-vectors/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">How to Find the Angle Between Two Vectors? </div> <div class="gfg-similar-read-item-subheading">Vector quantities are the physical quantities that have both magnitude and direction and the angle between two vectors can be easily found if the dot product or the cross product of the two vectors is given. In this article, we will learn how to find the angle between two vectors, its formula, relat</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/section-formula-vector-algebra/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Section Formula - Vector Algebra </div> <div class="gfg-similar-read-item-subheading">The section formula in vector algebra is a useful formula for finding a point that divides a line segment into a certain ratio. This concept is particularly important in physics and engineering for tasks such as finding center of mass, centroids, and other applications where a point must be located</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 11: Three-dimensional Geometry<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/direction-cosines-and-direction-ratios/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Direction Cosines and Direction Ratios </div> <div class="gfg-similar-read-item-subheading">Usually, for three-dimensional geometry, we rely on the three-dimensional Cartesian plane. Vectors can also be used to describe the lines and the angles they make with the axis. How should we describe a line passing through the origin making an angle with different axes? We define them using cosine</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/equation-of-a-line-in-3d/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Equation of a Line | Definition, Different Forms and Examples </div> <div class="gfg-similar-read-item-subheading">The equation of a line in a plane is given as y = mx + C where x and y are the coordinates of the plane, m is the slope of the line and C is the intercept. However, the construction of a line is not limited to a plane only. We know that a line is a path between two points. These two points can be lo</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/angles-between-two-lines-in-3d-space/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Angles Between two Lines in 3D Space | Solved Examples </div> <div class="gfg-similar-read-item-subheading">A line in mathematics and geometry is a fundamental concept representing a straight, one-dimensional figure that extends infinitely in both directions. Lines are characterized by having no thickness and being perfectly straight. Here are some important aspects and definitions related to lines: Key C</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/shortest-distance-between-two-lines-in-3d-space-class-12-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Shortest Distance Between Two Lines in 3D Space | Class 12 Maths </div> <div class="gfg-similar-read-item-subheading">The shortest distance between two lines in three-dimensional space is the length of the perpendicular segment drawn from a point on one line to the other line. 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We define a line as a geometrical figure that is extended in both direc</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">14 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 12: Linear Programming<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/linear-programming/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Linear Programming </div> <div class="gfg-similar-read-item-subheading">Linear programming is a mathematical concept that is used to find the optimal solution of the linear function. 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