# solid of revolution

Let $y=f(x)$ be a curve for $x$ in an interval $[a,b]$ satisfying $f(x)>0$ for $x$ in $(a,b)$. We may construct a corresponding solid of revolution, say $\mathcal{V}=\left\{(x,y,z):x\in[a,b]\mbox{ and }y^{2}+z^{2}\leq f\left(x\right% )^{2}\right\}$. Intuitively, it is the solid generated by rotating the surface $y\leq f(x)$ about the $x$-axis.

The interior of a surface of revolution is always a solid of revolution. These include

• the interior of a cylinder of radius $c>0$ and height $h$ with $f(x)=c$ for $0\leq x\leq h$,

• the interior of a sphere of radius $R>0$ with $f(x)=\sqrt{R^{2}-x^{2}}$ for $-R\leq x\leq R$, and

• the interior of a (right, circular) cone of base radius $R>0$ and height $h$ with $f(x)=Rx/h$ for $0\leq x\leq h$.

Let $\Gamma$ be a simple closed curve with parametrization $\alpha\left(t\right)=\left(X\left(t\right),Y\left(t\right)\right)$ for $t$ in an interval $[a,b]$ satisfying $Y\left(t\right)\geq 0$ for $t$ in $[a,b]$. By the Jordan curve theorem, we may choose the set of points, $\mathcal{S}$, ”inside” the curve, i.e. let $\mathcal{S}$ be the bounded connected component of the two connected components found in $\mathbb{R}^{2}\setminus\Gamma$. Another sort of solid of revolution is given by $\mathcal{V}=\left\{(x,y,z):x=X(t)\mbox{ for some }t\mbox{ in }[a,b]\mbox{ and % }y^{2}+z^{2}=s^{2}\mbox{ for some }s\mbox{ such that }(x,s)\in\mathcal{S}\cup% \Gamma\right\}$. Intuitively, it is the solid generated by rotating $\mathcal{S}\cup\Gamma$ about the $x$-axis.

Some examples of this sort of solid of revolution include

• the interior of a torus of minor radius $r>0$ and major radius $R>r$ with $\alpha\left(t\right)=\left(r\cos t,r\sin t+R\right)$ for $0\leq t\leq 2\pi$,

• a shell of a sphere with inner radius $r>0$ and outer radius $R>r$ with

 $\alpha\left(t\right)=\begin{cases}\left(R\cos\pi t,R\sin\pi t\right)&\mbox{ if% }t\in[0,1]\\ \left(r\left(1-t\right)+R\left(t-2\right),0\right)&\mbox{ if }t\in[1,2]\\ \left(-r\cos\pi t,r\sin\pi t\right)&\mbox{ if }t\in[2,3]\\ \left(r\left(4-t\right)+R\left(t-3\right),0\right)&\mbox{ if }t\in[3,4].\end{cases}$
Title solid of revolution SolidOfRevolution 2013-03-22 17:19:57 2013-03-22 17:19:57 nkirby (11104) nkirby (11104) 10 nkirby (11104) Definition msc 51M25 SurfaceOfRevolution2