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<p>
<span class="sidetocthetitle">MAT1001 Differential Calculus: Lecture Notes</span>
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<p>
Contents
</p>
</div>
<div class="sidetoccontents">
<p>
<a href="index.html" class="linkhome" >
Home</a>
</p>
<p>
<a href="Why-Do-We-Need-Calculus.html#autosec-5" class="tocchapter" >
<span class="sectionnumber">1</span> Why Do We Need Calculus</a>
</p>
<p>
<a href="Why-Do-We-Need-Calculus.html#autosec-6" class="tocsection" >
<span class="sectionnumber">1.1</span> Course Overview</a>
</p>
<p>
<a href="Why-Do-We-Need-Calculus.html#autosec-7" class="tocsection" >
<span class="sectionnumber">1.2</span> Why Calculus</a>
</p>
<p>
<a href="Functions.html#autosec-17" class="tocchapter" >
<span class="sectionnumber">2</span> Functions</a>
</p>
<p>
<a href="Functions.html#autosec-18" class="tocsection" >
<span class="sectionnumber">2.1</span> What is a function</a>
</p>
<p>
<a href="Functions.html#autosec-38" class="tocsection" >
<span class="sectionnumber">2.2</span> Trigonometric functions</a>
</p>
<p>
<a href="Functions.html#autosec-54" class="tocsection" >
<span class="sectionnumber">2.3</span> Logarithms and Exponentials</a>
</p>
<p>
<a href="Functions.html#autosec-65" class="tocsection" >
<span class="sectionnumber">2.4</span> Hyperbolic functions</a>
</p>
<p>
<a href="Functions.html#autosec-70" class="tocsection" >
<span class="sectionnumber">2.5</span> Limits and asymptotics</a>
</p>
<p>
<a href="Functions.html#autosec-84" class="tocsection" >
<span class="sectionnumber">2.6</span> Continuity and differentiability</a>
</p>
<p>
<a href="Differentiation.html#autosec-101" class="tocchapter" >
<span class="sectionnumber">3</span> Differentiation</a>
</p>
<p>
<a href="Differentiation.html#autosec-102" class="tocsection" >
<span class="sectionnumber">3.1</span> Differentiation from first principles</a>
</p>
<p>
<a href="Differentiation.html#autosec-111" class="tocsection" >
<span class="sectionnumber">3.2</span> Differentiation techniques</a>
</p>
<p>
<a href="Differentiation.html#autosec-131" class="tocsection" >
<span class="sectionnumber">3.3</span> Multiple derivatives</a>
</p>
<p>
<a href="Differentiation.html#autosec-133" class="tocsection" >
<span class="sectionnumber">3.4</span> Applications of differentiation</a>
</p>
<p>
<a href="Integration.html#autosec-157" class="tocchapter" >
<span class="sectionnumber">4</span> Integration</a>
</p>
<p>
<a href="Integration.html#autosec-158" class="tocsection" >
<span class="sectionnumber">4.1</span> Integration and antiderivatives</a>
</p>
<p>
<a href="Integration.html#autosec-168" class="tocsection" >
<span class="sectionnumber">4.2</span> Techniques for integration</a>
</p>
<p>
<a href="Integration.html#autosec-177" class="tocsection" >
<span class="sectionnumber">4.3</span> Definite integrals</a>
</p>
<p>
<a href="Integration.html#autosec-204" class="tocsection" >
<span class="sectionnumber">4.4</span> Applications of integrals</a>
</p>
<p>
<a href="Differential-Equations.html#autosec-216" class="tocchapter" >
<span class="sectionnumber">5</span> Differential Equations</a>
</p>
<p>
<a href="Numerical-Methods.html#autosec-218" class="tocchapter" >
<span class="sectionnumber">6</span> Numerical Methods</a>
</p>
<p>
<a href="Numerical-Methods.html#autosec-219" class="tocsection" >
<span class="sectionnumber">6.1</span> Solving equations numerically</a>
</p>
<p>
<a href="Numerical-Methods.html#autosec-251" class="tocsection" >
<span class="sectionnumber">6.2</span> Approximating functions</a>
</p>
<p>
<a href="Numerical-Methods.html#autosec-277" class="tocsection" >
<span class="sectionnumber">6.3</span> Numerical integration</a>
</p>
<p>
<a href="Numerical-Methods.html#autosec-303" class="tocsection" >
<span class="sectionnumber">6.4</span> Numerical differentiation</a>
</p>
<p>
<a href="Numerical-Methods.html#autosec-324" class="tocsection" >
<span class="sectionnumber">6.5</span> Numerical approaches to differential equations</a>
</p>
<p>
<a href="Calculus-in-Computer-Science.html#autosec-326" class="tocchapter" >
<span class="sectionnumber">7</span> Calculus in Computer Science</a>
</p>
<p>
<a href="Calculus-in-Computer-Science.html#autosec-327" class="tocsection" >
<span class="sectionnumber">7.1</span> LLMs and AI</a>
</p>
<p>
<a href="Calculus-in-Computer-Science.html#autosec-334" class="tocsection" >
<span class="sectionnumber">7.2</span> Robotics</a>
</p>
<p>
<a href="Advanced-Topics.html#autosec-336" class="tocchapter" >
<span class="sectionnumber">8</span> Advanced Topics</a>
</p>
<p>
<a href="Advanced-Topics.html#autosec-337" class="tocsection" >
<span class="sectionnumber">8.1</span> L’Hôpital’s rule for evaluating limits</a>
</p>
<p>
<a href="Advanced-Topics.html#autosec-341" class="tocsection" >
<span class="sectionnumber">8.2</span> Trigonometric Derivatives in Degrees</a>
</p>
<p>
<a href="Advanced-Topics.html#autosec-342" class="tocsection" >
<span class="sectionnumber">8.3</span> Functions of two variables</a>
</p>
<p>
<a href="Advanced-Topics.html#autosec-343" class="tocsection" >
<span class="sectionnumber">8.4</span> Multiple integrals</a>
</p>
<p>
<a href="Advanced-Topics.html#autosec-344" class="tocsection" >
<span class="sectionnumber">8.5</span> Optimisation Problems</a>
</p>
<p>
<a href="Advanced-Topics.html#autosec-347" class="tocsection" >
<span class="sectionnumber">8.6</span> Polynomial approximation</a>
</p>
<p>
<a href="Background-Mathematics.html#autosec-350" class="tocchapter" >
<span class="sectionnumber">9</span> Background Mathematics</a>
</p>
<p>
<a href="Background-Mathematics.html#autosec-351" class="tocsection" >
<span class="sectionnumber">9.1</span> Background and References</a>
</p>
<p>
<a href="Background-Mathematics.html#autosec-352" class="tocsection" >
<span class="sectionnumber">9.2</span> Polynomials and roots</a>
</p>
<p>
<a href="Background-Mathematics.html#autosec-354" class="tocsection" >
<span class="sectionnumber">9.3</span> Trigonometry Primer</a>
</p>
<p>
<a href="Background-Mathematics.html#autosec-358" class="tocsection" >
<span class="sectionnumber">9.4</span> Radians and Degrees</a>
</p>
<p>
<a href="Background-Mathematics.html#autosec-365" class="tocsection" >
<span class="sectionnumber">9.5</span> Rearranging Equations</a>
</p>
<p>
<a href="Background-Mathematics.html#autosec-370" class="tocsection" >
<span class="sectionnumber">9.6</span> More on Limits</a>
</p>
<p>
<a href="Further-Reading.html#autosec-373" class="tocchapter" >
<span class="sectionnumber">10</span> Further Reading</a>
</p>
<p>
<a href="Further-Reading.html#autosec-374" class="tocsection" >
<span class="sectionnumber">10.1</span> Why Calculus Extra Reading</a>
</p>
<p>
<a href="Further-Reading.html#autosec-375" class="tocsection" >
<span class="sectionnumber">10.2</span> Functions Extra Reading</a>
</p>
<p>
<a href="Further-Reading.html#autosec-376" class="tocsection" >
<span class="sectionnumber">10.3</span> Differentiation Extra Reading</a>
</p>
<p>
<a href="Further-Reading.html#autosec-377" class="tocsection" >
<span class="sectionnumber">10.4</span> Integration Extra Reading</a>
</p>
<p>
<a href="Further-Reading.html#autosec-378" class="tocsection" >
<span class="sectionnumber">10.5</span> Differential Equations Extra Reading</a>
</p>
<p>
<a href="Further-Reading.html#autosec-379" class="tocsection" >
<span class="sectionnumber">10.6</span> Numerical Methods Extra Reading</a>
</p>
<p>
<a href="Further-Reading.html#autosec-380" class="tocsection" >
<span class="sectionnumber">10.7</span> Calculus in Computer Science</a>
</p>
<p>
<a href="Further-Reading.html#autosec-381" class="tocsection" >
<span class="sectionnumber">10.8</span> Advanced Topics Extra Reading</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-383" class="tocchapter" >
<span class="sectionnumber">11</span> Tutorial Sheets</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-384" class="tocsection" >
<span class="sectionnumber">11.1</span> Week 1</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-400" class="tocsection" >
<span class="sectionnumber">11.2</span> Week 2</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-424" class="tocsection" >
<span class="sectionnumber">11.3</span> Week 3</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-442" class="tocsection" >
<span class="sectionnumber">11.4</span> Week 4</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-450" class="tocsection" >
<span class="sectionnumber">11.5</span> Week 5</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-473" class="tocsection" >
<span class="sectionnumber">11.6</span> Week 6</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-493" class="tocsection" >
<span class="sectionnumber">11.7</span> Week 7</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-518" class="tocsection" >
<span class="sectionnumber">11.8</span> Week 8</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-528" class="tocsection" >
<span class="sectionnumber">11.9</span> Week 9</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-540" class="tocsection" >
<span class="sectionnumber">11.10</span> Week 10</a>
</p>
<p>
<a href="Extra-Proofs-Derivations.html#autosec-551" class="tocchapter" >
<span class="sectionnumber">12</span> Extra Proofs and Derivations</a>
</p>
<p>
<a href="Standard-Derivatives-Integrals.html#autosec-554" class="tocchapter" >
<span class="sectionnumber">13</span> Standard Derivatives and Integrals</a>
</p>
<p>
<a href="Standard-Derivatives-Integrals.html#autosec-555" class="tocsection" >
<span class="sectionnumber">13.1</span> Derivatives</a>
</p>
<p>
<a href="Standard-Derivatives-Integrals.html#autosec-558" class="tocsection" >
<span class="sectionnumber">13.2</span> Integrals</a>
</p>
<p>
<a href="Bibliography.html#autosec-562" class="tocchapter" >
Bibliography</a>
</p>
</div>
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<main class="bodycontainer">
<section class="textbody">
<h1>MAT1001 Differential Calculus: Lecture Notes</h1>
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<h3 id="autosec-157">Chapter <span class="sectionnumber">4 </span>Integration</h3>
<a id="mat1001-autopage-157"></a>
<a id="mat1001-autofile-4"></a>
<a id="sec:integration"></a>
<div class="epigraph" role="note">
<div class="qitem">
<p>
<b>Integration</b>: The inverse process to differentiation, i.e. the process of finding a function with a derivative that is a given function.
</p>
<div class="epigraphsource">
<p>
<i>Penguin Dictionary of Mathematics</i>
</p>
</div>
</div>
</div>
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<h4 id="autosec-158"><span class="sectionnumber">4.1 </span>Integration and antiderivatives</h4>
<a id="mat1001-autopage-158"></a>
<p>
As the above quote from the <i>Penguin Dictionary of Mathematics</i> says, integration is the reverse process to differentiation. If differentiation is all about finding the rates of change of functions then we an think of integration as being about the areas under curves.
As with differentiation it is a process that was discovered to handle problems from physics so you can find plenty of examples of applications by looking into topics from mechanics. We may meet some of those at the end of this chapter, but for now we will focus on the
mathematics.<br />
</p>
<figure id="autoid-26" class="figure ">
<div class="center">
<p>
<span
id="lateximage-mat1001-19"
class="lateximagesource"
><!--
1 y
0.5
A1
x
π π 3π
2 2
−0.5 A2
−1
--><img
src="mat1001-images\image-19.svg"
alt="(Graph of sin(x).)"
style=""
class="lateximage"
></span>
</p>
<div class="figurecaption">
Figure 4.1: The graph of the sine function \(f(x)=\sin (x)\) with the area between the curve and the \(x\)-axis between \(x=0\) and \(x=3\uppi /2\) shaded. Note that the total shaded area is \(A=A_{1}+A_{2}\).
</div>
<a id="fig: sine function graph shaded"></a>
</div>
</figure>
<p>
The integral of a function over an interval is a measure of the signed area between the curve and the \(x\)-axis, with the convention that areas above the \(x\)-axis are positive and areas under the \(x\)-axis are negative. This is shown for \(f(x)=\sin (x)\) between
\(0\) and \(3\uppi /2\) in fig. <a href="Integration.html#fig: sine function graph shaded">4.1</a>. When we discuss numerical methods of integration in chapter <a href="Numerical-Methods.html#sec:numerics">6</a> we will understand how to
approximate these areas using thin rectangles as is shown in fig. <a href="Why-Do-We-Need-Calculus.html#fig: integral approximation">1.4</a>.<br />