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274 lines
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<!-- Mirrored from www.mathsisfun.com/calculus/integration-introduction.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:38:42 GMT -->
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<title>Introduction to Integration</title>
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<article id="content" role="main">
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<h1 class="center">Introduction to Integration</h1>
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<p class="center">Integration is a way of adding slices to find the whole.</p>
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<p>Integration can be used to find areas, volumes, central points and many useful things. But it is easiest to start with finding the <b>area between a function and the x-axis</b> like this:</p>
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<p class="center"><img src="images/integral-area.svg" alt="integral area = ?"><br>
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What is the area?</p>
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<h2>Slices</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>We could calculate the function at a few points and <b>add up slices of width <b>Δx</b></b> like this (but the answer won't be very accurate):</p></td>
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<td> </td>
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<td><img src="images/integral-area3.gif" alt="integral area big deltax" height="171" width="195"></td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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<td> </td>
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</tr>
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<td>
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<p>We can make <b>Δx</b> a lot smaller and <b>add up many small slices</b> (answer is getting better):</p>
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<p> </p></td>
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<td> </td>
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<td><img src="images/integral-area1.gif" alt="integral area small deltax" height="171" width="195"></td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td>
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<p>And as the slices <b>approach zero in width</b>, the answer approaches the <b>true answer</b>.</p>
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<p>We now write <b>dx</b> to mean the <b>Δx</b> slices are approaching zero in width.</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>That is a lot of adding up!</h2>
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<p>But we don't have to add them up, as there is a "shortcut", because ...</p>
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<p class="center larger">... finding an Integral is the <b>reverse</b> of finding a Derivative.</p>
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<p class="center">(So you should really know about <a href="derivatives-introduction.html">Derivatives</a> before reading more!)</p>
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<p>Like here:</p>
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<div class="example">
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<h3>Example: 2x</h3>
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<p class="center large">An integral of 2x is x<sup>2</sup> ...</p>
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<p class="center"><img src="images/integral-vs-derivative.svg" alt="integral vs derivative"></p>
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<p class="center larger">... because the derivative of x<sup>2</sup> is 2x</p>
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<p>
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(More about "+C" later.)
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</p>
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</div>
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<p>That simple example can be confirmed by calculating the area:</p>
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<p class="center"><img src="images/integral-2x-graph.svg" alt="integral 2x is x^2"></p>
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<p class="center large">Area of triangle = <span class="intbl"><em>1</em><strong>2</strong></span>(base)(height) = <span class="intbl"><em>1</em><strong>2</strong></span>(x)(2x) = x<sup>2</sup></p>
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<p>Integration can sometimes be that easy! </p>
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<h2>Notation</h2>
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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>
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<p>The symbol for "Integral" is a stylish "S"<br>
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(for "Sum", the idea of summing slices):</p></td>
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<td> </td>
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<td><img src="images/integral-notation-1.svg" alt="integral notation "></td>
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</tr>
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</tbody></table>
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<p>After the Integral Symbol we put the function we want to find the integral of (called the Integrand),</p>
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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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<p>And here is how we write the answer:</p>
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<p class="center"><img src="images/integral-notation-2.svg" alt="integral of 2x dx = x^2 + C"></p>
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<h2>Plus C</h2>
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<p>We wrote the answer as <span class="larger">x<sup>2</sup></span> but why <span class="larger">+C</span> ?</p>
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<p>It is the "Constant of Integration". It is there because of <b>all the functions whose derivative is 2x</b>:</p>
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<p class="center"><img src="images/integrals-vs-derivative.svg" alt="many integrals vs one derivative"></p>
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<ul>
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<li>the derivative of <b>x<sup>2</sup></b> is <b>2x</b>, </li>
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<li>and the derivative of <b>x<sup>2</sup>+4</b> is also <b>2x</b>, </li>
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<li>and the derivative of <b>x<sup>2</sup>+99</b> is also <b>2x</b>, </li>
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<li>and so on! </li></ul>
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<p>Because the derivative of a constant is zero.</p>
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<p>So when we <b>reverse</b> the operation (to find the integral) we only know <b>2x</b>, but there could have been a <b>constant of any value</b>.</p>
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<p>So we wrap up the idea by just writing <span class="larger">+ C</span> at the end.</p>
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<h2>A Practical Example: Tap and Tank</h2>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/integral-tap-tank.svg" alt="integral tap tank"></p>
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<p>Let us use a tap to fill a tank.</p>
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<p>The input (before integration) is the <b>flow rate</b> from the tap.</p>
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<p>We can integrate that flow (add up all the little bits of water) to give us the <b>volume of water</b> in the tank.</p>
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<div style="clear:both"></div>
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<p>Imagine a <b>Constant Flow Rate</b> of 1:</p>
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<p class="center"><img src="images/integral-tap-tank-constant.svg" alt="integral tap tank constant flow"></p>With a flow rate of <b>1</b>, the tank volume increases by <b>x</b>. That is <b>Integration</b>!
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<p class="center large">An integral of 1 is x</p>
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<div class="example">
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<p>With a flow rate of 1 liter per second, the volume increases by 1 liter every second, so would increase by 10 liters after 10 seconds, 60 liters after 60 seconds, etc.</p>
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<p>The flow rate stays at <b>1</b>, and the volume increases by <b>x</b></p></div>
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<p><i><b>And it works the other way too:</b></i></p>
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<p>If the tank volume increases by <b> x</b>, then the flow rate must be <b>1. <br></b></p>
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<div style="clear:both"></div>
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<p class="center large">The derivative of x is 1<br>
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</p><p>This shows that integrals and derivatives are opposites!</p>
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<p class="center"><img src="images/integral-vs-derivative-1-x.svg" alt="integral vs derivative"></p>
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<p> </p>
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<h3>Now For An Increasing Flow Rate</h3>
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<p>Imagine the flow starts at 0 and gradually increases (maybe a motor is slowly opening the tap):</p>
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<p class="center"><img src="images/integral-tap-tank-2x.svg" alt="integral tap tank flow rate 2x gives volume x^2"></p>
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<p>As the flow rate increases, the tank fills up faster and faster:</p>
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<ul>
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<li>Integration: With a flow rate of <b>2x</b>, the tank volume increases by <b>x<sup>2</sup></b></li>
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<li>Derivative: If the tank volume increases by <b>x<sup>2</sup></b>, then the flow rate must be <b>2x</b></li></ul>
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<p class="center"><img src="images/integral-vs-derivative.svg" alt="integral vs derivative"></p>
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<p>We can write it down this way:</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 style="text-align:right;">
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<p>The integral of the flow rate <b>2x</b> tells us the volume of water:</p>
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</td>
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<td> </td>
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<td nowrap="nowrap"><span class="center larger"><span style="font-size:150%;">∫</span>2x dx = x<sup>2</sup> + C</span></td>
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</tr>
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<tr>
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<td style="text-align:right;">
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<p>The derivative of the volume <b>x<sup>2</sup>+C</b> gives us back the flow rate:</p></td>
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<td> </td>
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<td nowrap="nowrap"><span class="center larger"><span class="intbl"><em>d</em><strong>dx</strong></span>(x<sup>2</sup> + C) = 2x</span></td>
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</tr>
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</tbody></table>
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<p> </p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/integral-tap-tank-graphs3.svg" alt="integral tap tank graphs"></p>
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<p>And hey, we even get a nice explanation of that "C" value ... maybe the tank already has water in it!</p>
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<ul>
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<li>The flow still increases the volume by the same amount</li>
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<li>And the increase in volume can give us back the flow rate.</li>
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</ul>
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<p>Which teaches us to always remember "+C".</p>
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<br><h2>Other functions</h2>
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<p>How do we integrate other functions?</p>
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<p>If we are lucky enough to find the function on the <b>result</b> side of a derivative, then (knowing that derivatives and integrals are opposites) we have an answer. But remember to add C.</p>
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<div class="example">
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<h3>Example: what is <span style="font-size:150%;">∫</span>cos(x) dx ?</h3>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/integral-vs-derivative-sin-cos.svg" alt="integral vs derivative, cos(x) vs sin(x)"></p>
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<p>From the <a href="derivatives-rules.html">Rules of Derivatives table</a> we see the derivative of sin(x) is cos(x) so:</p>
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<p class="center"><span class="larger"><span style="font-size:150%;">∫</span>cos(x) dx = sin(x) + C</span></p>
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</div>
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<p>But a lot of this "reversing" has already been done (see <a href="integration-rules.html">Rules of Integration</a>).</p>
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<div class="example">
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<h3>Example: What is <span style="font-size:150%;">∫</span>x<sup>3</sup> dx ?</h3>
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<p>On <a href="integration-rules.html">Rules of Integration</a> there is a "Power Rule" that says:</p>
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<p class="center larger"><span style="font-size:150%;">∫</span>x<sup>n</sup> dx = <span class="intbl"><em>x<sup>n+1</sup></em><strong>n+1</strong></span> + C</p>
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<p>We can use that rule with n=3:</p>
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<p class="center larger"><span style="font-size:150%;">∫</span>x<sup>3 </sup>dx = <span class="intbl"><em>x<sup>4</sup></em><strong>4</strong></span> + C</p>
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</div>
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<p class="larger">Knowing how to use those rules is the key to being good at Integration.</p>
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<p>So learn the rules and <b>get lots of practice</b>.</p>
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<div class="center80">
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<p class="center"><b>Learn the <a href="integration-rules.html">Rules of Integration</a> and Practice! Practice! Practice!</b><br>
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(there are some questions below to get you started)</p>
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</div>
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<h2>Definite vs Indefinite Integrals</h2>
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<p>We have been doing <b>Indefinite Integrals</b> so far.</p>
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<p>A <b>Definite Integral</b> has actual values to calculate between (they are put at the bottom and top of the "S"):</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/indefinite-integral.gif" alt="indefinite integral" height="210" width="195"></td>
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<td> </td>
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<td><img src="images/definite-integral.gif" alt="definite 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>Indefinite</b> Integral</td>
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<td> </td>
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<td><b>Definite</b> Integral</td>
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</tr>
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</tbody></table>
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<p>Read <a href="integration-definite.html">Definite Integrals</a> to learn more.</p>
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<p> </p>
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<div class="questions">
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<script>getQ(6824, 6825, 6826, 6827, 6828, 6829, 6830, 6831, 6832, 6833);</script>
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</div>
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<div class="related">
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<a href="integration-rules.html">Rules of Integration</a>
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<a href="derivative-vs-integral.html">Graphical Intro to Derivatives and Integrals</a>
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<a href="index.html">Calculus Index</a>
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</div>
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