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542 lines
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Definite Integrals</h1>
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<p class="center">You might like to read <a href="integration-introduction.html">Introduction to Integration</a> first!</p>
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<h2>Integration</h2>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td>
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<p><a href="integration-introduction.html">Integration</a> can be used to find areas, volumes, central points and many useful things. But it is often used to find the <b>area under the graph of a function</b> like this:</p></td>
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<td> </td>
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<td><span class="center"><img src="images/integral-area.gif" alt="integral area" height="171" width="195"></span></td>
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</tr>
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<tr>
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<td>
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<p>The area can be found by adding slices that <b>approach zero in width</b>:</p>
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<p>And there are <a href="integration-rules.html">Rules of Integration</a> that help us get the answer.</p></td>
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<td> </td>
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<td><img src="images/integral-area0.gif" alt="integral area dx" height="171" width="195"></td>
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</tr>
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</tbody></table>
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<h2>Notation</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/integral-notation-1.svg" alt="integral notation" height="121" width="306"></p>
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<p>The symbol for "Integral" is a stylish "S" (for "Sum", the idea of summing slices):</p>
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<p><br></p>After the Integral Symbol we put the function we want to find the integral of (called the Integrand).
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<p>And then finish with <b>dx</b> to mean the slices go in the x direction (and approach zero in width).</p>
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<h2>Definite Integral</h2>
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<p>A <b>Definite Integral</b> has start and end values: in other words there is an <b>interval</b> [a, b].</p>
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<p>a and b (called limits, bounds or boundaries) are put at the bottom and top of the "S", like this:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="images/definite-integral.gif" alt="definite integral" height="210" width="195"></td>
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<td style="width:30px;"> </td>
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<td><img src="images/indefinite-integral.gif" alt="indefinite integral" height="210" width="195"></td>
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</tr>
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<tr style="text-align:center;">
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<td><b>Definite</b> Integral<br>
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(from <b>a</b> to <b>b</b>)</td>
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<td> </td>
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<td><b>Indefinite</b> Integral<br>
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(no specific values)</td>
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</tr>
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</tbody></table>
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<p>We find the Definite Integral by calculating the <i>Indefinite</i> Integral at <b>a</b>, and at <b>b</b>, then subtracting:</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/integral-def-1graph.svg" alt="definite integral y=2x from 1 to 2 as graph" height="149" width="107"></p>
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<h3>Example: What is
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<div class="center" style="font-size:18px;">
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<div style="display:inline-block; text-align:center;">
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<div class="intto">2</div>
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<div class="intsymb">∫</div>
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<div class="intfrom">1</div>
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</div>
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<div class="inttext">2x dx</div>
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</div></h3>
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<p>We are being asked<span class="center"> for the <b>Definite Integral</b>, from 1 to 2, of <b>2x</b> dx</span></p>
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<p>First we need to find the <i><b>Indefinite</b></i><b> Integral</b>.</p>
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<p>Using the <a href="integration-rules.html">Rules of Integration</a> we find that <b><span style="font-size:150%;">∫</span>2x dx = x<sup>2</sup> + C</b></p>
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<p>Now calculate that at 1, and 2:</p>
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<ul>
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<li>At x=1: <span style="font-size:150%;">∫</span>2x dx = <b>1<sup>2</sup> + C</b></li>
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<li>At x=2: <span style="font-size:150%;">∫</span>2x dx = <b>2<sup>2</sup> + C</b></li>
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</ul>
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<p>Subtract:</p>
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<div class="so">(2<sup>2</sup> + C) − (1<sup>2</sup> + C) </div>
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<div class="so">2<sup>2</sup> + C − 1<sup>2</sup> − C </div>
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<div class="so">4 − 1 + <span class="hilite">C − C</span> = 3</div>
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<p>And "C" gets cancelled out ... so with <b>Definite Integrals we can ignore C</b>.</p>
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<p>Result:</p>
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<div class="center">
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<div style="display:inline-block; text-align:center;">
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<div class="intto">2</div>
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<div class="intsymb">∫</div>
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<div class="intfrom">1</div>
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</div>
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<div class="inttext">2x dx = <b>3</b></div>
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</div>
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<p> </p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/integral-def-12.svg" alt="area of y=2x from 1 to 2 equals 3" height="164" width="207"></p>
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<p><b>Check</b>: with such a simple shape, let's also try calculating the area by geometry:</p>
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<p class="center large">A = <span class="intbl"><em>2+4</em><strong>2</strong></span> × 1 = 3</p>
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<p>Yes, it does have an area of 3.</p>
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<p>(Yay!)</p>
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</div>
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<p><b>Notation</b>: It is usual to show the indefinite integral (without the +C) inside square brackets, with the limits <b>a</b> and <b>b</b> after, like this:</p>
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<div class="example">
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<h3>Example (continued)</h3>
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<p>How to show your answer:</p>
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<div class="tbl">
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<div class="row"><span class="left">
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<div style="display:inline-block; text-align:center;">
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<div class="intto">2</div>
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<div class="intsymb">∫</div>
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<div class="intfrom">1</div>
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</div>
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<div class="inttext">2x dx</div>
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</span><span class="rtlt" style="transform: translateY(-30%);">= <span style="font-size:140%;">[</span> x<sup>2</sup> <span style="font-size:140%;">]</span><div style="display:inline-block; text-align:center;transform: translateY(20%);">
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<div style="font-family: 'Times New Roman', Times, serif; ">2</div>
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<div style="font-family: 'Times New Roman', Times, serif; ">1</div>
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</div></span></div>
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<div class="row"><span class="left"> </span><span class="rtlt">= 2<sup>2</sup> − 1<sup>2</sup></span></div>
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<div class="row"><span class="left"> </span><span class="rtlt">= <b>3</b></span></div>
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</div>
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</div>
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<p> </p>
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<p>Let's try another example:</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/integral-def-5.svg" alt="definite integral y=cos(x) from 0.5 to 1 graph" height="119" width="147"></p>
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<h3>Example:</h3>
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<p class="center">The Definite Integral, from 0.5 to 1.0, of <b>cos(x)</b> dx:</p>
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<div class="center" style="font-size:18px;">
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<div style="display:inline-block; text-align:center;">
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<div class="intto">1</div>
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<div class="intsymb">∫</div>
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<div class="intfrom">0.5</div>
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</div>
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<div class="inttext">cos(x) dx</div>
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</div>
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<p>(Note: x must be in <a href="../geometry/radians.html">radians</a>)</p>
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<p> </p>
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<p>The <i>Indefinite</i> Integral is: <span style="font-size:150%;"><b>∫</b></span><b>cos(x) dx = sin(x) + C</b></p>
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<p>We can ignore C for definite integrals (as we saw above) and we get:</p>
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<div class="tbl">
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<div class="row"><span class="lt">
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<div style="display:inline-block; text-align:center;">
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<div class="intto">1</div>
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<div class="intsymb">∫</div>
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<div class="intfrom">0.5</div>
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</div>
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<div class="inttext">cos(x) dx</div></span><span class="rtlt" style="transform: translateY(-30%);">= <span style="font-size:140%;">[</span> sin(x) <span style="font-size:140%;">]</span>
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<div style="display:inline-block; text-align:center;transform: translateY(20%);">
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<div style="font-family: 'Times New Roman', Times, serif; ">1</div>
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<div style="font-family: 'Times New Roman', Times, serif; ">0.5</div>
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</div></span></div>
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<div class="row"><span class="left"> </span><span class="rtlt"> = sin(1) − sin(0.5)</span></div>
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<div class="row"><span class="left"> </span><span class="rtlt"> = 0.841... − 0.479...</span></div>
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<div class="row"><span class="left"> </span><span class="rtlt">= <b>0.362...</b></span></div>
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</div>
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</div>
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<p>And another example to make an important point:</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/integral-def-9.svg" alt="definite integral y=sin(x) from 0 to 1 graph" height="119" width="147"></p>
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<h3>Example:</h3>
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<p class="center">The Definite Integral, from 0 to 1, of <b>sin(x)</b> dx:</p>
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<div class="center" style="font-size:18px;">
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<div style="display:inline-block; text-align:center;">
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<div class="intto">1</div>
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<div class="intsymb">∫</div>
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<div class="intfrom">0</div>
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</div>
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<div class="inttext">sin(x) dx</div>
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</div>
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<p>The <i>Indefinite</i> Integral is: <span style="font-size:150%;"><b>∫</b></span><b>sin(x) dx = −cos(x) + C</b></p>
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<p>Since we are going from 0, <b>can we</b> just calculate the integral at x=1 ??</p>
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<p class="center">−cos(1) = −0.540...</p>
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<p>What? It is <b>negative</b>? But it looks positive in the graph.</p>
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<p>Well ... we made a <b>mistake</b>!</p>
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<p>Because <b>we need to subtract the integral at x=0</b>. We shouldn't assume it is zero.</p>
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<p> </p>
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<p>So let us do it properly, subtracting one from the other:</p>
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<div class="tbl">
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<div class="row"><span class="left">
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<div style="display:inline-block; text-align:center;">
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<div class="intto">1</div>
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<div class="intsymb">∫</div>
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<div class="intfrom">0</div>
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</div>
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<div class="inttext">sin(x) dx</div>
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</span><span class="rtlt" style="transform: translateY(-30%);">= <span style="font-size:140%;">[</span> −cos(x) <span style="font-size:140%;">]</span><div style="display:inline-block; text-align:center;transform: translateY(20%);">
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<div style="font-family: 'Times New Roman', Times, serif; ">1</div>
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<div style="font-family: 'Times New Roman', Times, serif; ">0</div>
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</div></span></div>
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<div class="row"><span class="left"> </span><span class="rtlt">= −cos(1) − (−cos(0))</span></div>
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<div class="row"><span class="left"> </span><span class="rtlt">= −0.540... − (−1)</span></div>
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<div class="row"><span class="left"> </span><span class="rtlt">= <b>0.460...</b> </span></div>
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</div>
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<p>That's better!</p>
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</div>
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<p>But we <i><b>can</b></i><b> have negative regions</b>, when the curve is below the axis:</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/integral-def-cos-1-3.svg" alt="definite integral y=cos(x) from 1 to 3 " height="127" width="212"></p>
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<h3>Example:</h3>
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<p class="center">The Definite Integral, from 1 to 3, of <b>cos(x)</b> dx:</p>
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<div class="center" style="font-size:18px;">
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<div style="display:inline-block; text-align:center;">
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<div class="intto">3</div>
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<div class="intsymb">∫</div>
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<div class="intfrom">1</div>
|
||
</div>
|
||
<div class="inttext">cos(x) dx</div>
|
||
</div>
|
||
|
||
<p class="center">Notice that some of it is positive, and some negative.<br>
|
||
The definite integral will work out the <b>net</b> value.</p>
|
||
<p> </p>
|
||
|
||
<p>Let us do the calculations:</p>
|
||
<div class="tbl">
|
||
|
||
<div class="row"><span class="left">
|
||
<div style="display:inline-block; text-align:center;">
|
||
<div class="intto">3</div>
|
||
<div class="intsymb">∫</div>
|
||
<div class="intfrom">1</div>
|
||
</div>
|
||
<div class="inttext">cos(x) dx</div>
|
||
</span>
|
||
|
||
<span class="rtlt" style="transform: translateY(-30%);">= <span style="font-size:140%;">[</span> sin(x) <span style="font-size:140%;">]</span><div style="display:inline-block; text-align:center;transform: translateY(20%);">
|
||
<div style="font-family: 'Times New Roman', Times, serif; ">3</div>
|
||
<div style="font-family: 'Times New Roman', Times, serif; ">1</div>
|
||
</div></span>
|
||
|
||
</div>
|
||
|
||
|
||
<div class="row"><span class="left">
|
||
|
||
</span> <span class="rtlt" style="vertical-align:middle;">= sin(3) − sin(1) </span></div>
|
||
<div class="row"><span class="left"> </span><span class="rtlt">= 0.141... − 0.841...</span></div>
|
||
<div class="row"><span class="left"> </span><span class="rtlt">= <b>−0.700...</b></span><span class="rtlt"></span></div>
|
||
</div>
|
||
<p>So there is more negative than positive with a net result of −0.700....</p>
|
||
</div>
|
||
So we have this important thing to remember:
|
||
<div class="def center">
|
||
<div style="display:inline-block; text-align:center;">
|
||
<div class="intto">b</div>
|
||
<div class="intsymb">∫</div>
|
||
<div class="intfrom">a</div>
|
||
</div>
|
||
<div class="inttext">f(x) dx = (Area above x axis) − (Area below x axis)</div>
|
||
|
||
</div>
|
||
<p> </p>
|
||
<p>Try integrating cos(x) with different start and end values to see for yourself how positives and negatives work.</p>
|
||
<h2>Positive Area</h2>
|
||
<p>But sometimes we want all area treated as <b>positive</b> (without the part below the axis being subtracted).</p>
|
||
<p>In that case we must calculate the areas <b>separately</b>, like in this example:</p>
|
||
<div class="example">
|
||
|
||
<span style="float:right; margin: 0 0 5px 10px;"><img src="images/integral-def-cos-1-3p.svg" alt="area y=cos(x) from 1 to 3 positive both above and below" height="127" width="212"></span>
|
||
<h3>Example: What is the <b>total area</b> between y = cos(x) and the x-axis, from x = 1 to x = 3?</h3>
|
||
<p>This is like the example we just did, but now we expect that <b>it is all positive</b> (imagine we had to paint it).</p>
|
||
<p>So now we have to do the parts separately:</p>
|
||
<ul>
|
||
<li>One for the area above the x-axis</li>
|
||
<li>One for the area below the x-axis</li>
|
||
</ul>
|
||
<p>The curve crosses the x-axis at x = <span class="times">π</span>/2 so we have:</p>
|
||
<p>From 1 to <span class="times">π</span>/2:</p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">
|
||
<div style="display:inline-block; text-align:center;">
|
||
<div class="intto"><span class="times">π</span>/2</div>
|
||
<div class="intsymb">∫</div>
|
||
<div class="intfrom">1</div>
|
||
</div>
|
||
<div class="inttext">cos(x) dx</div>
|
||
</span> <span class="rtlt" style="vertical-align:middle;">= sin(<span class="times">π</span>/2) − sin(1) </span></div>
|
||
<div class="row"><span class="left"> </span><span class="rtlt">= 1 − 0.841...</span></div>
|
||
<div class="row"><span class="left"> </span><span class="rtlt">= <b>0.158...</b></span></div>
|
||
</div>
|
||
<p>From <span class="times">π</span>/2 to 3:</p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">
|
||
<div style="display:inline-block; text-align:center;">
|
||
<div class="intto">3</div>
|
||
<div class="intsymb">∫</div>
|
||
<div class="intfrom"><span class="times">π</span>/2</div>
|
||
</div>
|
||
<div class="inttext">cos(x) dx</div>
|
||
</span> <span class="rtlt" style="vertical-align:middle;">= sin(3) − sin(<span class="times">π</span>/2) </span></div>
|
||
<div class="row"><span class="left"> </span><span class="rtlt">= 0.141... − 1</span></div>
|
||
<div class="row"><span class="left"> </span><span class="rtlt">= <b>−0.859...</b></span></div>
|
||
</div>
|
||
<div class="center">
|
||
<div style="display:inline-block; text-align:center;">
|
||
<div style="font-family: 'Times New Roman', Times, serif; "></div>
|
||
</div>
|
||
</div>
|
||
<p>That last one comes out negative, but we want it to be positive, so:</p>
|
||
<p class="center larger">Total area = 0.158... <span class="hilite">+</span> 0.859... = <b>1.017</b>...</p>
|
||
<p>This is very different from the answer in the previous example.</p>
|
||
</div>
|
||
<h2>Continuous</h2>
|
||
<p>Oh yes, the function we are integrating must be <a href="continuity.html">Continuous</a> between <b>a</b> and <b>b</b>: no holes, jumps or vertical asymptotes (where the function heads up/down towards infinity).</p>
|
||
<div class="example">
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/continuous-asymp-no.gif" alt="not continuous asymptote" height="109" width="147"></p>
|
||
<h3>Example:</h3>
|
||
<p>A vertical asymptote between <b>a</b> and <b>b</b> affects the definite integral.</p>
|
||
<div style="clear:both"></div>
|
||
</div>
|
||
<h2>Properties</h2>
|
||
<h3>Area above − area below</h3>
|
||
<p>The integral adds the area above the axis but subtracts the area below, for a "net value":</p>
|
||
<div class="center">
|
||
<div style="display:inline-block; text-align:center;">
|
||
<div class="intto">b</div>
|
||
<div class="intsymb">∫</div>
|
||
<div class="intfrom">a</div>
|
||
</div>
|
||
<div class="inttext">f(x) dx = (Area above x axis) − (Area below x axis)</div>
|
||
</div>
|
||
<p> </p>
|
||
<h3>Adding Functions</h3>
|
||
<p>The integral of <b>f+g</b> equals the integral of <b>f</b> plus the integral of <b>g</b>:</p>
|
||
<div class="center">
|
||
<div style="display:inline-block; text-align:center;">
|
||
<div class="intto">b</div>
|
||
<div class="intsymb">∫</div>
|
||
<div class="intfrom">a</div>
|
||
</div>
|
||
<div class="inttext">f(x) + g(x) dx = </div>
|
||
<div style="display:inline-block; text-align:center;">
|
||
<div class="intto">b</div>
|
||
<div class="intsymb">∫</div>
|
||
<div class="intfrom">a</div>
|
||
</div>
|
||
<div class="inttext">f(x) dx + </div>
|
||
<div style="display:inline-block; text-align:center;">
|
||
<div class="intto">b</div>
|
||
<div class="intsymb">∫</div>
|
||
<div class="intfrom">a</div>
|
||
</div>
|
||
<div class="inttext">g(x) dx </div>
|
||
</div>
|
||
<p> </p>
|
||
<h3>Reversing the interval</h3>
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/integral-def-propneg.svg" alt="definite integral negative property" height="94" width="142"></p>
|
||
<p>Reversing the direction of the interval gives the negative of the original direction.</p>
|
||
|
||
<div class="center">
|
||
<div style="display:inline-block; text-align:center;">
|
||
<div class="intto">b</div>
|
||
<div class="intsymb">∫</div>
|
||
<div class="intfrom">a</div>
|
||
</div>
|
||
<div class="inttext">f(x) dx = −</div>
|
||
<div style="display:inline-block; text-align:center;">
|
||
<div class="intto">a</div>
|
||
<div class="intsymb">∫</div>
|
||
<div class="intfrom">b</div>
|
||
</div>
|
||
<div class="inttext">f(x) dx</div>
|
||
|
||
</div>
|
||
|
||
|
||
|
||
<p> </p>
|
||
<h3>Interval of zero length</h3>
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/integral-def-prop0.svg" alt="definite integral area zero" height="94" width="142"></p>
|
||
<p>When the interval starts and ends at the same place, the result is zero:</p>
|
||
<div class="center">
|
||
<div style="display:inline-block; text-align:center;">
|
||
<div class="intto">a</div>
|
||
<div class="intsymb">∫</div>
|
||
<div class="intfrom">a</div>
|
||
</div>
|
||
<div class="inttext">f(x) dx = 0</div></div>
|
||
|
||
<h3>Adding intervals</h3>
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/integral-def-prop-add.svg" alt="area a to b = a to c plus c to b" height="94" width="142"></p>
|
||
<p>We can also add two adjacent intervals together:</p>
|
||
<div class="center">
|
||
<div style="display:inline-block; text-align:center;">
|
||
<div class="intto">b</div>
|
||
<div class="intsymb">∫</div>
|
||
<div class="intfrom">a</div>
|
||
</div>
|
||
<div class="inttext">f(x) dx = </div>
|
||
<div style="display:inline-block; text-align:center;">
|
||
<div class="intto">c</div>
|
||
<div class="intsymb">∫</div>
|
||
<div class="intfrom">a</div>
|
||
</div>
|
||
<div class="inttext">f(x) dx + </div>
|
||
<div style="display:inline-block; text-align:center;">
|
||
<div class="intto">b</div>
|
||
<div class="intsymb">∫</div>
|
||
<div class="intfrom">c</div>
|
||
</div>
|
||
<div class="inttext">f(x) dx </div>
|
||
</div>
|
||
|
||
<h2>Summary</h2>
|
||
<p>The Definite Integral between <b>a</b> and <b>b</b> is the Indefinite Integral at <b>b</b> minus the Indefinite Integral at<b> a</b>.</p>
|
||
<p> </p>
|
||
<div class="questions">6864, 6865, 6866, 6867, 6868, 6869, 6870, 6871, 6872, 6873, 6874</div>
|
||
|
||
<div class="related">
|
||
<a href="integration-introduction.html">Introduction to Integration</a>
|
||
<a href="index.html">Calculus Index</a>
|
||
</div>
|
||
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