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