Mathematics · Ch 9 — Applications of Integration

Introduction

9.1

Introduction

Note

"Give me a place to stand and I will move the earth" — Archimedes.

One of the earliest mathematicians to compute areas and volumes of geometric shapes rigorously was Archimedes of Syracuse (287 BCE–212 BCE), a Greek mathematician, physicist, engineer and inventor. Archimedes proved that the area enclosed between a parabola and a chord (straight line) is exactly $\dfrac43$ times the area of the triangle inscribed in that parabolic segment (with the same base and the vertex where the tangent is parallel to the base). He arrived at this by segmenting the region into infinitely many elementary strips and summing them — the very idea of a limiting sum that lies at the heart of the definite integral this chapter develops, and which we then apply to compute areas and volumes of geometric shapes.

Recap: the anti-derivative. If a function $F(x)$ can be found such that $\dfrac{d}{dx}F(x)=f(x)$ , then $F(x)$ is called an anti-derivative (or indefinite integral) of $f(x)$ . It is never unique: for any constant $C$ , $\dfrac{d}{dx}[F(x)+C]=f(x)$ too, so all anti-derivatives of a given $f$ differ only by a constant. The anti-derivative is written $\displaystyle\int f(x)\,dx$ .

Linearity of the indefinite integral. For constants $\alpha,\beta$ ,

\int\big[\alpha f(x)+\beta g(x)\big]\,dx=\alpha\int f(x)\,dx+\beta\int g(x)\,dx.

A working table of standard indefinite integrals (each verified by differentiating the right-hand side), used freely throughout the chapter:

$f(x)$	$\displaystyle\int f(x)\,dx$
$K$ (constant)	$Kx+C$
$(ax+b)^n,\ a\ne0,\ n\ne-1$	$\dfrac{1}{a(n+1)}(ax+b)^{n+1}+C$
$\dfrac{1}{ax+b},\ a\ne0$	$\dfrac1a\log_e
$e^{ax}$	$\dfrac{e^{ax}}{a}+C$
$\sin(ax+b)$	$-\dfrac{\cos(ax+b)}{a}+C$
$\cos(ax+b)$	$\dfrac{\sin(ax+b)}{a}+C$
$\tan(ax+b)$	$\dfrac1a\log
$\cot(ax+b)$	$\dfrac1a\log
$\sec(ax+b)$	$\dfrac1a\log
$\operatorname{cosec}(ax+b)$	$-\dfrac1a\log
$\dfrac{1}{a^2+x^2}$	$\dfrac1a\tan^{-1}\dfrac xa+C$
$\dfrac{1}{a^2-x^2}$	$\dfrac{1}{2a}\log_e\left
$\dfrac{1}{x^2-a^2}$	$\dfrac{1}{2a}\log_e\left
$\dfrac{1}{\sqrt{a^2-x^2}}$	$\sin^{-1}\dfrac xa+C$
$\dfrac{1}{\sqrt{x^2+a^2}}$	$\log_e\left(x+\sqrt{x^2+a^2}\right)+C$
$\dfrac{1}{\sqrt{x^2-a^2}}$	$\log_e\left
$\sqrt{a^2+x^2}$	$\dfrac x2\sqrt{a^2+x^2}+\dfrac{a^2}{2}\log_e\left(x+\sqrt{a^2+x^2}\right)+C$
$\sqrt{a^2-x^2}$	$\dfrac x2\sqrt{a^2-x^2}+\dfrac{a^2}{2}\sin^{-1}\dfrac xa+C$
$\sqrt{x^2-a^2}$	$\dfrac x2\sqrt{x^2-a^2}-\dfrac{a^2}{2}\log_e\left

What this chapter builds toward. Starting from the geometric, sum-based idea of a definite integral (§9.2), the chapter reaches the Fundamental Theorems (§9.3) that make evaluation practical, adds computational tools — Bernoulli's formula (§9.4), improper integrals (§9.5), reduction formulae (§9.6) and the gamma integral (§9.7) — and finally applies all of this machinery to the geometric payoff: computing areas of plane regions (§9.8) and volumes of solids of revolution (§9.9), fulfilling Archimedes' original programme with full rigour.

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