Calculus I
&
Engineering Mathematics 2
Workshop 8
Density and Mass
If a rod has variable density $\rho(x)$ along its length, the mass is obtained by integration.
$\ds m = \int_a^b \rho(x)\,dx.$
Here $\rho(x)$ represents mass per unit length.
Area Between Two Curves
The area between two curves $y=f(x)$ and $y=g(x)$ over $[a,b]$ is
$\ds A = \int_a^b \big(f(x)-g(x)\big)\,dx,$
where $f(x)$ is the upper curve and $g(x)$ the lower curve.
Volume of Revolution
When a region is revolved about an axis, its volume can be computed using integration:
$\ds V = \pi \int_a^b \big[f(x)\big]^2\,dx.$
This is the disk method for rotation about the $x$-axis.
Volume of Revolution
Improper Integrals
An integral is called improper if the interval is infinite or the integrand is unbounded:
$\ds \int_1^{\infty} \frac{1}{x^2}\,dx$ $=\ds \lim_{b\to\infty} \int_1^b \frac{1}{x^2}\,dx$ $=\ds \lim_{b\to\infty}\left[-\frac{1}{x}\right]_1^b\,dx$
$=\ds \lim_{b\to\infty}\left[-\frac{1}{b}+1\right]$ $=\ds 1$
Note: The integral converges if the limit exists and is finite.
Applications of Integration — Summary
| Application | Recognise | Formula |
|---|---|---|
| Density / Mass | $\rho(x)$ along an interval $[a,b]$ | $\ds m=\int_a^b \rho(x)\,dx$ |
| Area between curves | $y=f(x),\ y=g(x),\,$ $f(x)\geq g(x)$ | $\ds A=\int_a^b \big(f(x)-g(x)\big)\,dx$ |
| Volume of revolution | Rotation about an axis | $\ds V=\pi\int_a^b \big[f(x)\big]^2\,dx$ |
| Improper integrals |
$\infty$ limit or vertical asymptote |
$\ds \int_a^\infty f(x)\,dx=\lim_{b\to\infty}\int_a^b f(x)\,dx$ |