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 $𝒱=\{(x,y,z):x\in [a,b]\text{ and }{y}^2+z^2\le f{\left(x\right)}^2\}$. Intuitively, it is the solid generated by rotating the surface $y\le f(x)$ 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 $0\le x\le h$,

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

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

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

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