PlanetMath: solid of revolution

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solid of revolution

(Definition)

Let be a curve for in an interval satisfying for in . 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 -axis.

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

the interior of a cylinder of radius and height with for $0\leq x\leq h$ ,
the interior of a sphere of radius 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 and height with 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 in an interval satisfying $Y\left(t\right)\geq 0$ for in . 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 $\mathcal{V}=\left \{ (x,y,z): x=X(t) \mbox{ for some } t \mbox{ in } [a,b] \mb... ...box{ for some } s \mbox{ such that } (x,s)\in \mathcal{S} \cup \Gamma \right \}$ . Intuitively, it is the solid generated by rotating $\mathcal{S}\cup \Gamma$ about the -axis.

Some examples of this sort of solid of revolution include

the interior of a torus of minor radius and major radius 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 and outer radius with
$\displaystyle \alpha\left(t\right)=\begin{cases}\left(R\cos \pi t, R\sin \pi t\... ...ft(4-t\right)+R\left(t-3\right),0\right) & \mbox{ if } t \in [3,4]. \end{cases}$