Applications of Integration

Applications of Integration — Formula Sheet

1. Definite integral as a limit of a sum

With \(h=\dfrac{b-a}{n}\),

\[\int_a^b f(x)\,dx=\lim_{n\to\infty}h\sum_{r=1}^{n}f(a+rh)=\lim_{n\to\infty}\frac{b-a}{n}\sum_{r=1}^{n}f\left(a+r\,\frac{b-a}{n}\right).\]

Useful special case \((a=0,\ b=1)\):

\[\int_0^1 f(x)\,dx=\lim_{n\to\infty}\frac{1}{n}\sum_{r=1}^{n}f\left(\frac{r}{n}\right).\]

2. Fundamental Theorem of Calculus

First form: if \(F(x)=\displaystyle\int_a^x f(t)\,dt\), then \(F'(x)=f(x)\).

Second form (evaluation): if \(F'(x)=f(x)\), then \(\displaystyle\int_a^b f(x)\,dx=F(b)-F(a)=\big[F(x)\big]_a^b.\)

3. Properties of definite integrals

\(\displaystyle\int_a^b f(x)\,dx=\int_a^b f(t)\,dt\) (the variable is a dummy).
\(\displaystyle\int_a^b f(x)\,dx=-\int_b^a f(x)\,dx\).
\(\displaystyle\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx,\quad a
\(\displaystyle\int_a^b f(x)\,dx=\int_a^b f(a+b-x)\,dx\).
\(\displaystyle\int_0^a f(x)\,dx=\int_0^a f(a-x)\,dx\) (the "king" property).
\(\displaystyle\int_0^{2a} f(x)\,dx=\int_0^a\big[f(x)+f(2a-x)\big]\,dx\); hence \(2\displaystyle\int_0^a f(x)\,dx\) if \(f(2a-x)=f(x)\), and \(0\) if \(f(2a-x)=-f(x)\).
Symmetry on \([-a,a]\): \(\displaystyle\int_{-a}^{a} f(x)\,dx=2\int_0^a f(x)\,dx\) if \(f\) is even, and \(0\) if \(f\) is odd.

4. Bernoulli's formula (generalised integration by parts)

For \(\displaystyle\int u\,v\,dx\) with \(u\) a polynomial, writing \(v_1=\int v\,dx,\ v_2=\int v_1\,dx,\dots\):

\[\int u\,v\,dx=uv_1-u'v_2+u''v_3-u'''v_4+\cdots\]

The series terminates because the successive derivatives of a polynomial eventually vanish.

5. Improper integrals and the Gamma function

\[\Gamma(n)=\int_0^\infty e^{-x}x^{\,n-1}\,dx,\qquad n>0.\]

\(\Gamma(n+1)=n\,\Gamma(n),\quad \Gamma(1)=1,\quad \Gamma(n+1)=n!\) for a non-negative integer \(n\).
\(\displaystyle\int_0^\infty e^{-ax}x^{\,n-1}\,dx=\frac{\Gamma(n)}{a^{\,n}}\quad(a>0)\).
\(\displaystyle\int_0^\infty e^{-ax}x^{\,n}\,dx=\frac{n!}{a^{\,n+1}}\quad(a>0,\ n=0,1,2,\dots)\).

6. Reduction / Wallis formulae

\[\int_0^{\pi/2}\sin^{n}x\,dx=\int_0^{\pi/2}\cos^{n}x\,dx= \begin{cases} \dfrac{(n-1)(n-3)\cdots 2}{\,n(n-2)\cdots 3\,}, & n\ \text{odd},\\[3mm] \dfrac{(n-1)(n-3)\cdots 1}{\,n(n-2)\cdots 2\,}\cdot\dfrac{\pi}{2}, & n\ \text{even}. \end{cases}\]

For mixed powers with \(m,n\) positive integers,

\[\int_0^{\pi/2}\sin^{m}x\,\cos^{n}x\,dx=\frac{\big[(m-1)(m-3)\cdots\big]\big[(n-1)(n-3)\cdots\big]}{(m+n)(m+n-2)\cdots}\times k,\]

where \(k=\dfrac{\pi}{2}\) when \(m\) and \(n\) are both even, and \(k=1\) otherwise.

7. Area of a plane region

Above the \(x\)-axis, between \(x=a\) and \(x=b\): \(\displaystyle A=\int_a^b y\,dx\).
Below the \(x\)-axis: \(\displaystyle A=\left|\int_a^b y\,dx\right|=-\int_a^b y\,dx\).
Right of the \(y\)-axis, between \(y=c\) and \(y=d\): \(\displaystyle A=\int_c^d x\,dy\).
Left of the \(y\)-axis: \(\displaystyle A=\left|\int_c^d x\,dy\right|\).
Between two curves with \(y_U\ge y_L\): \(\displaystyle A=\int_a^b\big(y_U-y_L\big)\,dx\).
Between two curves with \(x_R\ge x_L\): \(\displaystyle A=\int_c^d\big(x_R-x_L\big)\,dy\).

8. Volume of a solid of revolution (disc method)

About the \(x\)-axis: \(\displaystyle V=\pi\int_a^b y^{2}\,dx\).
About the \(y\)-axis: \(\displaystyle V=\pi\int_c^d x^{2}\,dy\).

9. Standard results worth memorising

Sphere of radius \(a\): \(V=\dfrac{4}{3}\pi a^{3}\).
Right circular cone, base radius \(r\), height \(h\): \(V=\dfrac{1}{3}\pi r^{2}h\).
Area of the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\) is \(\pi a b\); circle of radius \(a\) is \(\pi a^{2}\).

Applications of Integration — Formula Sheet