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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

\begin{align}2\pi\int\limits_a^b f(x)\sqrt{1+f'(x)^2}dx\end{align}

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.

\begin{align} V&=2\pi\iint_R y\,dA=2\pi\int\limits_{-r}^r\int\limits_0^{\sqrt{r^2-x^2}}y\,dy\,dx=2\pi\int\limits_{-r}^r\left[\frac{y^2}{2}\right]_0^{\sqrt{r^2-x^2}}\\ &=\pi\int\limits_{-r}^r(r^2-x^2)dx=\pi\left[r^2x-\frac{x^3}{3}\right]_{-r}^r=\frac{4\pi}{3}r^3 \end{align}

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