Volume of Revolution

The volume of revolution formed when $y=f\left(x\right)$ is rotated about the x-axis between a and b is given by:

$$V=\mathrm{\pi }{\int }_a^b{y}^{2}dx$$

The volume of revolution formed when $x=f\left(y\right)$ is rotated about the y-axis between $y=a$ and $y=b$ is given by:

$$V=\mathrm{\pi }{\int }_a^b{x}^{2}dy$$

The following standard formulae may help when adding and subtracting volumes:

• Volume of a cylinder of height, h and radius, r is $\mathrm{\pi }{r}^{2}h$;
• Volume of a cone of height h and base radius r is $V=\frac{1}{3}\mathrm{\pi }{r}^{2}h$.