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A Solid of revolution is a solid formed by the rotation of a function around a line. Many common shapes, such as spheres, cones, and cylinders are solids of revolution. The volume of such a solid can be calculated by using rings or shells, or by using a double integral in the form

$2 \pi \iint_R (x-b) dy dx$

assuming the rotation is about the line y=b.

The surface of revolution is the surface enclosing the solid. The surface area of a surface of revolution can be found with the formula

$2\pi \int_{a}^{b}f(x)\sqrt{1 + f'(x)^2}dx$

## Examples

Find the volume of the solid of revolution obtained when the function

$f(x) = \sqrt{r^2 - x^2}$

(a semicircle rotated to obtain a sphere) is rotated about the x-axis.

$V = 2 \pi \iint_R y dA = 2 \pi \int_{-r}^{r} \int_0^{\sqrt{r^2 - x^2}} y dy dx = 2 \pi \int_{-r}^r [\frac{y^2}{2}]_0^{\sqrt{r^2 - x^2}}$
$= \pi \int_{-r}^r (r^2 - x^2) dx = \pi [r^2 x - \frac{x^3}{3}]_{-r}^{r} = \frac{4}{3} \pi r^3$