Skip to main content
\(\require{mathrsfs}\require{cancel}\newcommand{\dee}[1]{\mathrm{d}#1} \newcommand{\half}{ \frac{1}{2} } \newcommand{\ds}{\displaystyle} \newcommand{\ts}{\textstyle} \newcommand{\es}{ {\varnothing}} \newcommand{\st}{ {\mbox{ s.t. }} } \newcommand{\pow}[1]{ \mathcal{P}\left(#1\right) } \newcommand{\set}[1]{ \left\{#1\right\} } \newcommand{\lin}{{\text{LIN}}} \newcommand{\quot}{{\text{QR}}} \newcommand{\simp}{{\text{SMP}}} \newcommand{\diff}[2]{ \frac{\mathrm{d}#1}{\mathrm{d}#2}} \newcommand{\bdiff}[2]{ \frac{\mathrm{d}}{\mathrm{d}#2} \left( #1 \right)} \newcommand{\ddiff}[3]{ \frac{\mathrm{d}^#1#2}{\mathrm{d}{#3}^#1}} \renewcommand{\neg}{ {\sim} } \newcommand{\limp}{ {\;\Rightarrow\;} } \newcommand{\nimp}{ {\;\not\Rightarrow\;} } \newcommand{\liff}{ {\;\Leftrightarrow\;} } \newcommand{\niff}{ {\;\not\Leftrightarrow\;} } \newcommand{\De}{\Delta} \newcommand{\bbbn}{\mathbb{N}} \newcommand{\bbbr}{\mathbb{R}} \newcommand{\bbbp}{\mathbb{P}} \newcommand{\cI}{\mathcal{I}} \newcommand{\cR}{\mathcal{R}} \newcommand{\cV}{\mathcal{V}} \newcommand{\Si}{\Sigma} \newcommand{\arccsc}{\mathop{\mathrm{arccsc}}} \newcommand{\arcsec}{\mathop{\mathrm{arcsec}}} \newcommand{\arccot}{\mathop{\mathrm{arccot}}} \newcommand{\erf}{\mathop{\mathrm{erf}}} \newcommand{\smsum}{\mathop{{\ts \sum}}} \newcommand{\atp}[2]{ \genfrac{}{}{0in}{}{#1}{#2} } \newcommand{\ave}{\mathrm{ave}} \newcommand{\llt}{\left \lt } \newcommand{\rgt}{\right \gt } \newcommand{\YEaxis}[2]{\draw[help lines] (-#1,0)--(#1,0) node[right]{$x$};\draw[help lines] (0,-#2)--(0,#2) node[above]{$y$};} \newcommand{\YEaaxis}[4]{\draw[help lines] (-#1,0)--(#2,0) node[right]{$x$};\draw[help lines] (0,-#3)--(0,#4) node[above]{$y$};} \newcommand{\YEtaxis}[4]{\draw[help lines] (-#1,0)--(#2,0) node[right]{$t$};\draw[help lines] (0,-#3)--(0,#4) node[above]{$y$};} \newcommand{\YEtaaxis}[4]{\draw[help lines, <->] (-#1,0)--(#2,0) node[right]{$t$}; \draw[help lines, <->] (0,-#3)--(0,#4) node[above]{$y$};} \newcommand{\YExcoord}[2]{\draw (#1,.2)--(#1,-.2) node[below]{$#2$};} \newcommand{\YEycoord}[2]{\draw (.2,#1)--(-.2,#1) node[left]{$#2$};} \newcommand{\YEnxcoord}[2]{\draw (#1,-.2)--(#1,.2) node[above]{$#2$};} \newcommand{\YEnycoord}[2]{\draw (-.2,#1)--(.2,#1) node[right]{$#2$};} \newcommand{\YEstickfig}[3]{ \draw (#1,#2) arc(-90:270:2mm); \draw (#1,#2)--(#1,#2-.5) (#1-.25,#2-.75)--(#1,#2-.5)--(#1+.25,#2-.75) (#1-.2,#2-.2)--(#1+.2,#2-.2);} \newcommand{\IBP}[7]{ \begin{array}{|c | l | l |} \hline \color{red}{\text{Option 1:}} & u=#2 &\color{red}{\dee{u}=#3 ~ \dee{#1}} \\ & \dee{v}=#5~\dee{#1} &\color{red}{v=#7} \\ \hline \color{blue}{\text{Option 2:}} & u=#5 &\color{blue}{\dee{u}=#6 ~ \dee{#1}} \\ &\dee{v}=#2 \dee{#1} &\color{blue}{v=#4} \\ \hline \end{array} } \renewcommand{\textcolor}[2]{{\color{#1}{#2}}} \newcommand{\trigtri}[4]{ \begin{tikzpicture} \draw (-.5,0)--(2,0)--(2,1.5)--cycle; \draw (1.8,0) |- (2,.2); \draw[double] (0,0) arc(0:30:.5cm); \draw (0,.2) node[right]{$#1$}; \draw (1,-.5) node{$#2$}; \draw (2,.75) node[right]{$#3$}; \draw (.6,1.1) node[rotate=30]{$#4$}; \end{tikzpicture}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Chapter1Integration

Calculus is built on two operations — differentiation and integration.

  • Differentiation — as we saw last term, differentiation allows us to compute and study the instantaneous rate of change of quantities. At its most basic it allows us to compute tangent lines and velocities, but it also led us to quite sophisticated applications including approximation of functions through Taylor polynomials and optimisation of quantities by studying critical and singular points.
  • Integration — at its most basic, allows us to analyse the area under a curve. Of course, its application and importance extend far beyond areas and it plays a central role in solving differential equations.

<<SVG image is unavailable, or your browser cannot render it>>

It is not immediately obvious that these two topics are related to each other. However, as we shall see, they are indeed intimately linked.