Applications of Integration — Formula Sheet

Applications of Integration — Formula Sheet

Definite integral as a limit of a sum

\[ \int_{a}^{b} f(x)\,dx = \lim_{n\to\infty}\frac{b-a}{n}\sum_{i=1}^{n} f\!\left(a+i\,\frac{b-a}{n}\right). \]

Fundamental Theorems of Integral Calculus

\[ \frac{d}{dx}\int_{a}^{x} f(u)\,du = f(x), \qquad \int_{a}^{b} f(x)\,dx = F(b)-F(a)\ \text{ where } F'=f. \]

Properties of definite integrals

\[ \int_{a}^{b} f(x)\,dx = \int_{a}^{b} f(a+b-x)\,dx, \qquad \int_{0}^{a} f(x)\,dx = \int_{0}^{a} f(a-x)\,dx. \] \[ \int_{-a}^{a} f(x)\,dx = \begin{cases} 2\displaystyle\int_{0}^{a} f(x)\,dx, & f \text{ even},\\[1mm] 0, & f \text{ odd}.\end{cases} \] \[ \int_{0}^{2a} f(x)\,dx = \begin{cases} 2\displaystyle\int_{0}^{a} f(x)\,dx, & f(2a-x)=f(x),\\[1mm] 0, & f(2a-x)=-f(x).\end{cases} \]

Standard inverse-function integrals

\[ \int \frac{dx}{\sqrt{a^{2}-x^{2}}} = \sin^{-1}\frac{x}{a}+C, \qquad \int \frac{dx}{a^{2}+x^{2}} = \frac{1}{a}\tan^{-1}\frac{x}{a}+C. \]

Bernoulli's formula (integration by parts, repeated)

\[ \int u\,v\,dx = u v_{1} - u' v_{2} + u'' v_{3} - \cdots, \]

where \(u',u'',\dots\) are successive derivatives of the polynomial \(u\), and \(v_{1},v_{2},\dots\) are successive integrals of \(v\).

Reduction (Wallis) formulae

\[ \int_{0}^{\pi/2}\sin^{n}x\,dx=\int_{0}^{\pi/2}\cos^{n}x\,dx = \begin{cases}\dfrac{(n-1)(n-3)\cdots 1}{n(n-2)\cdots 2}\cdot\dfrac{\pi}{2}, & n \text{ even},\\[2mm] \dfrac{(n-1)(n-3)\cdots 2}{n(n-2)\cdots 3}, & n \text{ odd}.\end{cases} \] \[ \int_{0}^{1} x^{m}(1-x)^{n}\,dx = \frac{m!\,n!}{(m+n+1)!}. \]

Gamma integral

\[ \Gamma(n) = \int_{0}^{\infty} e^{-x} x^{\,n-1}\,dx = (n-1)!, \qquad \Gamma(n+1)=n\,\Gamma(n), \] \[ \int_{0}^{\infty} e^{-ax}\,x^{n}\,dx = \frac{n!}{a^{\,n+1}}\quad (a>0). \]

Area of a bounded plane region

\[ A=\int_{a}^{b} y\,dx \ \ (\text{above x-axis}), \qquad A=\left|\int_{a}^{b} y\,dx\right| \ \ (\text{below x-axis}), \] \[ A=\int_{c}^{d} x\,dy \ \ (\text{right of y-axis}), \qquad A=\left|\int_{c}^{d} x\,dy\right| \ \ (\text{left of y-axis}). \] \[ A=\int_{a}^{b}\big(y_{U}-y_{L}\big)\,dx \quad (f\ge g), \qquad A=\int_{c}^{d}\big(x_{R}-x_{L}\big)\,dy. \]

Volume of a solid of revolution

\[ V=\pi\int_{a}^{b} y^{2}\,dx \ \ (\text{about x-axis}), \qquad V=\pi\int_{c}^{d} x^{2}\,dy \ \ (\text{about y-axis}). \]