Calculus | Definition & Facts

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class="md-content"> <div class="md-article-container template-desktop"> <div class="infinite-scroll-container article last"> <article class="article-content container-lg qa-content px-0 pt-0 pb-40 py-lg-20 content md-expanded" data-topic-id="89161"> <div class="grid gx-0"> <div class="col-auto"> <div class="topic-left-rail md-article-drawer position-relative d-flex border-right-sm border-left-sm open"> <div class="drawer d-flex flex-column open"> <div class="left-rail-section-content"> <div class="topic-left-rail-header text-truncate bg-gray-50 position-relative text-right d-flex align-items-center"> <div class="tlr-title px-20 py-15 text-left"> <em class="material-icons text-gray-400 d-lg-none" data-icon="toc"></em> <a class="font-serif font-weight-bold text-black link-blue" href="https://www.britannica.com/science/calculus-mathematics">calculus</a> </div> <button aria-label="Close" class="js-sections-close-button btn-link btn-sm btn d-lg-none position-absolute top-0 p-10 right-0" > <em class="material-icons font-26" data-icon="close"></em> </button> </div> <div class="section-content pl-10 pr-20 pl-sm-50 pr-sm-60 pl-lg-5 pr-lg-10 pt-10 pt-lg-0 bg-gray-50 clear-catfish-ad"> <div class="toc mb-20"> <div class="font-serif font-14 font-weight-bold mx-15 mb-15 mt-20"> Table of Contents </div> <ul class="list-unstyled my-0" data-level="h1"><li data-target="#ref1"><div class="pl-25"><a class="link-gray-900 w-100" href="/science/calculus-mathematics">Introduction</a></div><div class="ml-40 toc-drawer sub-toc-drawer"></div></li><li data-target="#ref252423"><div class="d-flex align-items-center"><div class="ml-25"></div><a class="w-100 link-gray-900" href="/science/calculus-mathematics#ref252423">Calculating curves and areas under curves</a></div><div class="ml-40 toc-drawer sub-toc-drawer"></div></li><li data-target="#ref252424"><div class="d-flex align-items-center"><div class="ml-25"></div><a class="w-100 link-gray-900" href="/science/calculus-mathematics#ref252424">Calculating velocities and slopes</a></div><div class="ml-40 toc-drawer sub-toc-drawer"></div></li><li data-target="#ref252425"><div class="d-flex align-items-center"><div class="ml-25"></div><a class="w-100 link-gray-900" href="/science/calculus-mathematics#ref252425">Differentiation and integration</a></div><div class="ml-40 toc-drawer sub-toc-drawer"></div></li></ul> <a class="toc-extra-link link-gray-900" href="https://www.britannica.com/science/calculus-mathematics/additional-info">References & Edit History</a> <a class="toc-extra-link link-gray-900" href="/facts/calculus-mathematics">Related Topics</a> </div> <div class="tlr-media-slider pb-10 mb-30"> <a class="section-header link-gray-900 font-serif font-14 font-weight-bold mb-10 mx-10" href="https://www.britannica.com/science/calculus-mathematics/images-videos">Images</a> <div class="slider js-slider position-relative d-inline-flex align-items-center mw-100 "> <div 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As the time interval h approaches zero, the secant (average speed) approaches the tangent (actual, or instantaneous, speed) at (t, f(t))." height="50" /> </a> <a href="https://cdn.britannica.com/78/26978-004-9E41D387/region-transformation-formula-constant-area-circumference-rectangle.jpg" data-href="/media/1/89161/19628" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/78/26978-004-9E41D387/region-transformation-formula-constant-area-circumference-rectangle.jpg" alt="The transformation of a circular region into an approximately rectangular regionThis suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr2. As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr2—the same area as that of the circle. 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As the time interval h approaches zero, the secant (average speed) approaches the tangent (actual, or instantaneous, speed) at (t, f(t))." loading="eager" /> </picture> <button class="magnifying-glass btn btn-circle position-absolute shadow btn-white top-10 right-10" aria-label="Zoom in"> <em class="material-icons link-blue" data-icon="zoom_in"></em> </button> </a> </div> <figcaption class="card-body"> <div class="md-assembly-caption text-muted font-14 font-serif line-clamp"> <span> <a class="gtm-assembly-link md-assembly-title font-weight-bold d-inline font-sans-serif mr-5 media-overlay-link" href="https://cdn.britannica.com/82/26982-004-92145859/difference-illustration-secant-rates-average-change-f.jpg" data-href="/media/1/89161/19632"></a> <span><span>An illustration of the difference between average and instantaneous rates of changeThe graph of <em>f</em>(<em>t</em>) shows the secant between (<em>t</em>, <em>f</em>(<em>t</em>)) and (<em>t</em> + <em>h</em>, <em>f</em>(<em>t</em> + <em>h</em>)) and the tangent to <em>f</em>(<em>t</em>) at <em>t</em>. As the time interval <em>h</em> approaches zero, the secant (average speed) approaches the tangent (actual, or instantaneous, speed) at (<em>t</em>, <em>f</em>(<em>t</em>)).</span></span> <button class="js-more-btn d-none btn btn-unstyled font-12 bg-white js-content" aria-label="Toggle more/less fact data"> <span class="link-blue">(more)</span> </button> </span> </div> </figcaption> </figure> </div><div class="topic-header"> <div class="d-flex align-items-top justify-content-between"> <div class="d-flex flex-column"> <div> <div> <h1>calculus</h1></div> </div> <div class="topic-identifier font-16 font-md-20">mathematics</div> </div> </div> <div class="d-none d-sm-flex flex-row"> <div class="mr-10 mb-15"> <button class="ai-ask-button btn border-2 btn-sm js-inline-ai-ask-button btn-outline-red-400 border-red-400"> Ask the Chatbot a Question </button> </div> <div class="d-none d-sm-block md-topic-tools qa-action-buttons mb-15" data-topic-id="89161"> <button class="js-tooltip btn btn-sm btn-outline-blue border pr-10 border-2" > <em class="material-icons md-icon ml-n10 my-n5 mr-5" data-icon="more_vert"></em> More Actions </button> <div class="md-more-popover popover popover-sm p-0 font-14 z-1"> <div> <button class="js-print-modal-button js-modal gtm-topic-tool btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" data-modal="[data-topic-id=89161] .md-print-modal" > <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="print"></em> Print </button> <div class="md-print-modal size-lg d-none"> <div class="md-modal-body"> <div class="h2 font-serif d-flex align-items-center pb-15 border-bottom"> <em class="material-icons text-blue mr-10">print</em> Print </div> <div class="mt-20 mb-10"> Please select which sections you would like to print: </div> <form action="/print/article/89161" method="post" target="_blank" rel="noopener"> <div class="print-box-items"> <ul class="list-unstyled"> <li><label><input class="mr-10" type="checkbox" name="sequence[]" value="0">Table Of Contents</label></li> </ul> </div> <input type="submit" class="btn btn-blue md-disabled" value="Print" /> </form> </div> </div> </div> <div> <button class="js-modal qa-cite-modal-button btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" data-modal="[data-topic-id=89161] .md-cite-modal"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="verified"></em> Cite </button> <div class="md-cite-modal size-lg d-none"> <div class="md-modal-body"> <div class="h2 font-serif d-flex align-items-center pb-15 border-bottom mb-15"> <em class="material-icons text-blue mr-10">verified</em>Cite </div> <div class="font-serif"> While every effort has been made to follow citation style rules, there may be some discrepancies. 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They write new content and verify and edit content received from contributors.</div> </a> <div data-popper-arrow></div> </div> <span class="btn btn-link editor-link p-0 qa-byline-link font-12 "> The Editors of Encyclopaedia Britannica</span></div> <div class="last-updated font-12 font-serif"> <span class="text-gray-700"> Last Updated: <time datetime="2024-10-30T00:00:00CDT" >Oct 30, 2024</time> •</span> <a class="byline-edit-history" href="https://www.britannica.com/science/calculus-mathematics/additional-info#history" rel="nofollow">Article History</a> </div></div> </div> <button class="d-flex d-lg-none btn btn-outline-blue border rounded-sm shadow-sm mobile-toc-button gtm-mobile-toc-inline-button d-none d-sm-block js-sections-inline-button module-spacing btn d-lg-none"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="toc"></em> Table of Contents </button> <div class="d-flex d-sm-none flex-row"> <button class="d-flex d-lg-none btn btn-outline-blue border rounded-sm shadow-sm mobile-toc-button gtm-mobile-toc-inline-button js-sections-inline-button module-spacing"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="toc"></em> Table of Contents </button> <button class="ai-ask-button btn border-2 ai-ask-button btn border-2 module-spacing btn-sm js-inline-ai-ask-button btn-outline-red-400 border-red-400 p-10 ml-5"> Ask the Chatbot a Question </button> </div> <div class="js-qf-module qf-module px-40 px-sm-20 py-15 mx-auto module-spacing font-14 bg-gray-50 rounded"> <div class="facts-list mt-10"> <div class=""> <div class="js-fact mb-10 line-clamp clamp-3"> <dl> <dt>Key People: </dt> <dd><a href="/biography/Isaac-Newton" topicid="413189">Isaac Newton</a></dd> <dd><a href="/biography/Leonhard-Euler" topicid="195201">Leonhard Euler</a></dd> <dd><a href="/biography/John-Wallis" topicid="634927">John Wallis</a></dd> <dd><a href="/biography/Augustin-Louis-Baron-Cauchy" topicid="100302">Augustin-Louis Cauchy</a></dd> <dd><a href="/biography/Johann-Bernoulli" topicid="62606">Johann Bernoulli</a></dd> </dl> <button class="js-more-btn d-none btn btn-unstyled font-12 bg-gray-50" aria-label="Toggle more/less fact data"> <em class="js-content link-blue">(Show more)</em> </button> </div> </div> <div class=""> <div class="js-fact mb-10 line-clamp clamp-3"> <dl> <dt>Related Topics: </dt> <dd><a href="/science/differentiation-mathematics" topicid="162982">differentiation</a></dd> <dd><a href="/science/integration-mathematics" topicid="289677">integration</a></dd> <dd><a href="/science/integral-calculus" topicid="1521439">integral calculus</a></dd> <dd><a href="/science/fundamental-theorem-of-calculus" topicid="222221">fundamental theorem of calculus</a></dd> <dd><a href="/science/differential-calculus" topicid="1516642">differential calculus</a></dd> </dl> <button class="js-more-btn d-none btn btn-unstyled font-12 bg-gray-50" aria-label="Toggle more/less fact data"> <em class="js-content link-blue">(Show more)</em> </button> </div> <div class="text-center"> <a class="btn btn-sm btn-link p-0" href="/facts/calculus-mathematics"> See all related content </a> </div> </div> </div> </div><span class="marker before-article"></span><section data-level="1" id="ref1"><span class="marker PREMOD1 mod-inline"></span><p class="topic-paragraph"><strong><span id="ref67401"></span>calculus</strong>, branch of <a href="https://www.britannica.com/science/mathematics" class="md-crosslink autoxref " data-show-preview="true">mathematics</a> concerned with the calculation of instantaneous rates of change (<a href="https://www.britannica.com/science/analysis-mathematics/Calculus#ref218268" class="md-crosslink " data-show-preview="true">differential calculus</a>) and the summation of infinitely many small factors to determine some whole (<a href="https://www.britannica.com/science/analysis-mathematics/Higher-order-derivatives#ref218274" class="md-crosslink " data-show-preview="true">integral calculus</a>). Two mathematicians, <span id="ref67402"></span><a href="https://www.britannica.com/biography/Isaac-Newton" class="md-crosslink " data-show-preview="true">Isaac Newton</a> of England and <span id="ref67403"></span><a href="https://www.britannica.com/biography/Gottfried-Wilhelm-Leibniz" class="md-crosslink " data-show-preview="true">Gottfried Wilhelm Leibniz</a> of Germany, share credit for having independently developed the calculus in the 17th century. Calculus is now the basic entry point for anyone wishing to study <a href="https://www.britannica.com/science/physics-science" class="md-crosslink autoxref " data-show-preview="true">physics</a>, chemistry, biology, <a href="https://www.britannica.com/money/economics" class="md-crosslink autoxref " data-show-preview="true">economics</a>, finance, or actuarial <a href="https://www.britannica.com/science/science" class="md-crosslink autoxref " data-show-preview="true">science</a>. Calculus makes it possible to solve problems as <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="diverse" href="https://www.merriam-webster.com/dictionary/diverse" data-type="MW">diverse</a> as tracking the position of a <a href="https://www.britannica.com/technology/space-shuttle" class="md-crosslink autoxref " data-show-preview="true">space shuttle</a> or predicting the <a href="https://www.britannica.com/science/pressure" class="md-crosslink autoxref " data-show-preview="true">pressure</a> building up behind a dam as the water rises. Computers have become a valuable tool for solving calculus problems that were once considered impossibly difficult.</p><span class="marker MOD1 mod-inline"></span></section> <span class="marker h2"></span><section data-level="1" id="ref252423"><h2 class="h1">Calculating curves and areas under curves</h2> <span class="marker PREMOD2 mod-inline"></span><p class="topic-paragraph">The roots of calculus lie in some of the oldest <a href="https://www.britannica.com/science/geometry" class="md-crosslink " data-show-preview="true">geometry</a> problems on record. The Egyptian <a href="https://www.britannica.com/topic/Rhind-papyrus" class="md-crosslink " data-show-preview="true">Rhind papyrus</a> (<em>c.</em> 1650 <span class="text-smallcaps">bce</span>) gives rules for finding the <span id="ref67406"></span><a href="https://www.britannica.com/science/area" class="md-crosslink ">area</a> of a circle and the volume of a truncated pyramid. Ancient Greek geometers investigated finding tangents to curves, the <a href="https://www.britannica.com/science/centre-of-gravity" class="md-crosslink autoxref " data-show-preview="true">centre of gravity</a> of plane and solid figures, and the volumes of objects formed by revolving various curves about a fixed axis.</p><span class="marker MOD2 mod-inline"></span> <span class="marker PREMOD3 mod-inline"></span><p class="topic-paragraph">By 1635 the Italian mathematician <span id="ref891984"></span><a href="https://www.britannica.com/biography/Bonaventura-Cavalieri" class="md-crosslink " data-show-preview="true">Bonaventura Cavalieri</a> had supplemented the rigorous tools of Greek geometry with <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="heuristic" href="https://www.merriam-webster.com/dictionary/heuristic" data-type="MW">heuristic</a> methods that used the idea of infinitely small segments of lines, areas, and volumes. In 1637 the French mathematician-philosopher <span id="ref891985"></span><a href="https://www.britannica.com/biography/Rene-Descartes" class="md-crosslink " data-show-preview="true">René Descartes</a> published his invention of <a href="https://www.britannica.com/science/analytic-geometry" class="md-crosslink " data-show-preview="true">analytic geometry</a> for giving algebraic descriptions of geometric figures. Descartes’s method, in combination with an ancient idea of curves being generated by a moving point, allowed mathematicians such as Newton to describe <a href="https://www.britannica.com/science/motion-mechanics" class="md-crosslink autoxref " data-show-preview="true">motion</a> algebraically. Suddenly geometers could go beyond the single cases and ad hoc methods of previous times. They could see patterns of results, and so conjecture new results, that the older geometric language had obscured.</p><a class="link-module shadow-sm d-block qa-quiz-module" href="/quiz/Numbers-and-mathematics" data-link-module-iframe-link=""> <img loading="lazy" src="https://cdn.britannica.com/86/94086-131-0BAE374D/Equations-blackboard.jpg" alt="Equations written on blackboard" class="rounded-sm mr-15" width="70" /> <div class="line-clamp clamp-5"> <div class="module-title bg-green">Britannica Quiz</div> <div class="font-weight-semi-bold mt-5">Numbers and Mathematics</div> </div> </a><span class="marker MOD3 mod-inline"></span> <span class="marker PREMOD4 mod-inline"></span><p class="topic-paragraph">For example, the Greek geometer <span id="ref67405"></span><a href="https://www.britannica.com/biography/Archimedes" class="md-crosslink " data-show-preview="true">Archimedes</a> (287–212/211 <span class="text-smallcaps">bce</span>) discovered as an isolated result that the area of a segment of a <a href="https://www.britannica.com/science/parabola" class="md-crosslink " data-show-preview="true">parabola</a> is equal to a certain triangle. But with algebraic notation, in which a parabola is written as <em>y</em> = <em>x</em><sup>2</sup>, Cavalieri and other geometers soon noted that the area between this curve and the <em>x</em>-axis from 0 to <em>a</em> is <em>a</em><sup>3</sup>/3 and that a similar rule holds for the curve <em>y</em> = <em>x</em><sup>3</sup>—namely, that the corresponding area is <em>a</em><sup>4</sup>/4. From here it was not difficult for them to guess that the general formula for the area under a curve <em>y</em> = <em>x</em><sup><em>n</em></sup> is <em>a</em><sup><em>n</em> + 1</sup>/(<em>n</em> + 1).</p><span class="marker MOD4 mod-inline"></span></section> <span class="marker h3"></span><section data-level="1" id="ref252424"><h2 class="h1">Calculating velocities and slopes</h2> <span class="marker PREMOD5 mod-inline"></span><p class="topic-paragraph">The problem of finding <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="tangents" href="https://www.britannica.com/dictionary/tangents" data-type="EB">tangents</a> to curves was closely related to an important problem that arose from the Italian scientist <a href="https://www.britannica.com/biography/Galileo-Galilei" class="md-crosslink autoxref " data-show-preview="true">Galileo Galilei’s</a> investigations of motion, that of finding the <span id="ref67407"></span><a href="https://www.britannica.com/science/velocity" class="md-crosslink " data-show-preview="true">velocity</a> at any instant of a particle moving according to some law. Galileo established that in <em>t</em> seconds a freely falling body falls a distance <em>g</em><em>t</em><sup>2</sup>/2, where <em>g</em> is a <a href="https://www.britannica.com/topic/constant" class="md-crosslink autoxref " data-show-preview="true">constant</a> (later interpreted by Newton as the <a href="https://www.britannica.com/science/gravity-physics" class="md-crosslink " data-show-preview="true">gravitational</a> constant). With the definition of average velocity as the distance per time, the body’s average velocity over an interval from <em>t</em> to <em>t</em> + <em>h</em> is given by the expression [<em>g</em>(<em>t</em> + <em>h</em>)<sup>2</sup>/2 − <em>g</em><em>t</em><sup>2</sup>/2]/<em>h</em>. This simplifies to <em>g</em><em>t</em> + <em>g</em><em>h</em>/2 and is called the <span id="ref892019"></span>difference quotient of the <a href="https://www.britannica.com/science/function-mathematics" class="md-crosslink autoxref " data-show-preview="true">function</a> <em>g</em><em>t</em><sup>2</sup>/2. As <em>h</em> approaches 0, this formula approaches <em>g</em><em>t</em>, which is interpreted as the instantaneous velocity of a falling body at time <em>t</em>.</p><span class="marker MOD5 mod-inline"></span> <span class="marker PREMOD6 mod-inline"></span><p class="topic-paragraph">This expression for motion is identical to that obtained for the <a href="https://www.britannica.com/science/slope-mathematics" class="md-crosslink autoxref " data-show-preview="true">slope</a> of the <a href="https://www.britannica.com/science/tangent-of-a-curve" class="md-crosslink autoxref " data-show-preview="true">tangent</a> to the parabola <em>f</em>(<em>t</em>) = <em>y</em> = <em>g</em><em>t</em><sup>2</sup>/2 at the point <em>t</em>. In this geometric <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="context" href="https://www.merriam-webster.com/dictionary/context" data-type="MW">context</a>, the expression <em>g</em><em>t</em> + <em>g</em><em>h</em>/2 (or its equivalent [<em>f</em>(<em>t</em> + <em>h</em>) − <em>f</em>(<em>t</em>)]/<em>h</em>) denotes the slope of a secant <a href="https://www.britannica.com/science/line-mathematics" class="md-crosslink autoxref " data-show-preview="true">line</a> connecting the point (<em>t</em>, <em>f</em>(<em>t</em>)) to the nearby point (<em>t</em> + <em>h</em>, <em>f</em>(<em>t</em> + <em>h</em>)) (<em>see</em> <span class="link-blue media-overlay-link asmref" data-href="/media/1/89161/19632">figure</span>). In the <a href="https://www.britannica.com/science/limit-mathematics" class="md-crosslink " data-show-preview="true">limit</a>, with smaller and smaller intervals <em>h</em>, the secant line approaches the tangent line and its slope at the point <em>t</em>.</p><span class="marker MOD6 mod-inline"></span> <span class="marker PREMOD7 mod-inline"></span><p class="topic-paragraph">Thus, the difference quotient can be interpreted as instantaneous velocity or as the slope of a tangent to a curve. It was the calculus that established this deep connection between geometry and physics—in the process transforming physics and giving a new <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="impetus" href="https://www.merriam-webster.com/dictionary/impetus" data-type="MW">impetus</a> to the study of geometry.</p><div class="module-spacing"> <DIV class="marketing-INLINE_SUBSCRIPTION marketing-content" data-marketing-id="INLINE_SUBSCRIPTION"><style> .student-promo-banner-wrapper { container-type: inline-size; margin-bottom: 15px; } @container (min-width: 475px) { .student-promo-banner { flex-direction: row; } .student-promo-banner-img-wrapper { margin-bottom: 0; margin-right: 10px; justify-content: flex-start; } .student-promo-banner-text-wrapper { text-align: left; margin-bottom: 0px; margin-left: 10px; } .student-promo-banner-button-wrapper { margin-right: 0; } }</style> <div class="student-promo-banner-wrapper"> <div class="student-promo-banner d-flex flex-column align-items-center bg-blue rounded p-20"> <div class="student-promo-banner-img-wrapper mb-20 mr-0 d-flex justify-content-center"> <img class="rounded" style="max-width: 100px; min-width: 80px" src="https://cdn.britannica.com/marketing/BlueThistle.webp" /> </div> <div class="student-promo-banner-text-wrapper ml-0 mb-10 text-center text-white"> <div class="h2 mb-10">Get Unlimited Access</div> <div class="h4 font-weight-semi-bold">Try Britannica Premium for free and discover more.</div> </div> <div class="student-promo-banner-button-wrapper d-flex justify-content-center align-items-center ml-auto mr-auto"> <a class="btn btn-m btn-orange" href="https://premium.britannica.com/premium-membership/?utm_source=premium&utm_medium=inline-cta&utm_campaign=black-friday-2024">Subscribe</a> </div> </div> </div> </DIV></div><span class="marker MOD7 mod-inline"></span></section> <span class="marker h4"></span><section data-level="1" id="ref252425"><h2 class="h1">Differentiation and integration</h2> <span class="marker PREMOD8 mod-inline"></span><p class="topic-paragraph">Independently, Newton and Leibniz established simple rules for finding the formula for the slope of the tangent to a curve at any point on it, given only a formula for the curve. The rate of change of a function <em>f</em> (denoted by <em>f</em>′) is known as its <span id="ref67408"></span><a href="https://www.britannica.com/science/derivative-mathematics" class="md-crosslink " data-show-preview="true">derivative</a>. Finding the formula of the derivative function is called <span id="ref67412"></span><a href="https://www.britannica.com/science/differentiation-mathematics" class="md-crosslink " data-show-preview="true">differentiation</a>, and the rules for doing so form the basis of differential calculus. Depending on the context, derivatives may be interpreted as slopes of tangent lines, velocities of moving particles, or other quantities, and therein lies the <a href="https://www.britannica.com/topic/great-power" class="md-crosslink autoxref " data-show-preview="true">great power</a> of the <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="differential" href="https://www.britannica.com/dictionary/differential" data-type="EB">differential</a> calculus.</p><span class="marker MOD8 mod-inline"></span> <span class="marker PREMOD9 mod-inline"></span><p class="topic-paragraph">An important application of differential calculus is graphing a curve given its <a href="https://www.britannica.com/science/equation" class="md-crosslink autoxref " data-show-preview="true">equation</a> <em>y</em> = <em>f</em>(<em>x</em>). This involves, in particular, finding local maximum and minimum points on the <a href="https://www.britannica.com/science/graph-mathematics" class="md-crosslink autoxref " data-show-preview="true">graph</a>, as well as changes in inflection (convex to concave, or vice versa). When examining a function used in a <a href="https://www.britannica.com/science/mathematical-model" class="md-crosslink " data-show-preview="true">mathematical model</a>, such geometric notions have physical interpretations that allow a scientist or engineer to quickly gain a feeling for the behaviour of a physical system.</p><span class="marker MOD9 mod-inline"></span> <span class="marker PREMOD10 mod-inline"></span><p class="topic-paragraph">The other great discovery of Newton and Leibniz was that finding the derivatives of functions was, in a precise sense, the <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="inverse" href="https://www.britannica.com/dictionary/inverse" data-type="EB">inverse</a> of the problem of finding areas under curves—a principle now known as the <span id="ref892020"></span><a href="https://www.britannica.com/science/analysis-mathematics/Higher-order-derivatives#ref218275" class="md-crosslink " data-show-preview="true">fundamental theorem of calculus</a>. Specifically, Newton discovered that if there exists a function <em>F</em>(<em>t</em>) that denotes the area under the curve <em>y</em> = <em>f</em>(<em>x</em>) from, say, 0 to <em>t</em>, then this function’s derivative will equal the original curve over that interval, <em>F</em>′(<em>t</em>) = <em>f</em>(<em>t</em>). Hence, to find the area under the curve <em>y</em> = <em>x</em><sup>2</sup> from 0 to <em>t</em>, it is enough to find a function <em>F</em> so that <em>F</em>′(<em>t</em>) = <em>t</em><sup>2</sup>. The differential calculus shows that the most general such function is <em>x</em><sup>3</sup>/3 + <em>C</em>, where <em>C</em> is an arbitrary constant. This is called the <span id="ref67413"></span><a href="https://www.britannica.com/topic/indefinite-integral" class="md-crosslink ">(indefinite)</a> <a href="https://www.britannica.com/science/integral-mathematics" class="md-crosslink " data-show-preview="true">integral</a> of the function <em>y</em> = <em>x</em><sup>2</sup>, and it is written as ∫<em>x</em><sup>2</sup><em>d</em><em>x</em>. The initial symbol ∫ is an elongated S, which stands for sum, and <em>d</em><em>x</em> indicates an infinitely small increment of the variable, or axis, over which the function is being summed. Leibniz introduced this because he thought of <span id="ref250561"></span><a href="https://www.britannica.com/science/integration-mathematics" class="md-crosslink " data-show-preview="true">integration</a> as finding the area under a curve by a summation of the areas of infinitely many <a href="https://www.britannica.com/science/infinitesimal" class="md-crosslink " data-show-preview="true">infinitesimally</a> thin rectangles between the <em>x</em>-axis and the curve. Newton and Leibniz discovered that <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="integrating" href="https://www.merriam-webster.com/dictionary/integrating" data-type="MW">integrating</a> <em>f</em>(<em>x</em>) is equivalent to solving a <a href="https://www.britannica.com/science/differential-equation" class="md-crosslink " data-show-preview="true">differential equation</a>—i.e., finding a function <em>F</em>(<em>t</em>) so that <em>F</em>′(<em>t</em>) = <em>f</em>(<em>t</em>). In physical terms, solving this equation can be interpreted as finding the distance <em>F</em>(<em>t</em>) traveled by an object whose velocity has a given expression <em>f</em>(<em>t</em>).</p><div class="one-good-fact-module"> </div><span class="marker MOD10 mod-inline"></span> <span class="marker PREMOD11 mod-inline"></span><p class="topic-paragraph">The branch of the calculus concerned with calculating <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="integrals" href="https://www.merriam-webster.com/dictionary/integrals" data-type="MW">integrals</a> is the <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="integral" href="https://www.merriam-webster.com/dictionary/integral" data-type="MW">integral</a> calculus, and among its many applications are finding <a href="https://www.britannica.com/science/work-physics" class="md-crosslink autoxref " data-show-preview="true">work</a> done by physical systems and calculating pressure behind a dam at a given depth.</p><span class="marker MOD11 mod-inline"></span><span class="md-signature"><a href="/contributor/John-L-Berggren/3484">John L. 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Calculus | Definition & Facts | Britannica