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 $\mathcal{V}=\{(x,y,z):x\in [a,b]\text{and}{y}^{2}+{z}^{2}\le f{\left(x\right)}^{2}\}$. 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

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the interior of a cylinder^{} of radius $c>0$ and height $h$ with $f(x)=c$ for $0\le x\le h$,

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

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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, $\mathcal{S}$, ”inside” the curve, i.e. let $\mathcal{S}$ be the bounded^{} connected component^{} of the two connected components found in ${\mathbb{R}}^{2}\setminus \mathrm{\Gamma}$. Another sort of solid of revolution is given by $\mathcal{V}=\{(x,y,z):x=X(t)\text{for some}t\text{in}[a,b]\text{and}{y}^{2}+{z}^{2}={s}^{2}\text{for some}s\text{such that}(x,s)\in \mathcal{S}\cup \mathrm{\Gamma}\}$. Intuitively, it is the solid generated by rotating $\mathcal{S}\cup \mathrm{\Gamma}$ about the $x$axis.
Some examples of this sort of solid of revolution include

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

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a shell of a sphere with inner radius $r>0$ and outer radius $R>r$ with
$$\alpha \left(t\right)=\{\begin{array}{cc}(R\mathrm{cos}\pi t,R\mathrm{sin}\pi t)\hfill & \text{if}t\in [0,1]\hfill \\ (r\left(1t\right)+R\left(t2\right),0)\hfill & \text{if}t\in [1,2]\hfill \\ (r\mathrm{cos}\pi t,r\mathrm{sin}\pi t)\hfill & \text{if}t\in [2,3]\hfill \\ (r\left(4t\right)+R\left(t3\right),0)\hfill & \text{if}t\in [3,4].\hfill \end{array}$$
Title  solid of revolution 

Canonical name  SolidOfRevolution 
Date of creation  20130322 17:19:57 
Last modified on  20130322 17:19:57 
Owner  nkirby (11104) 
Last modified by  nkirby (11104) 
Numerical id  10 
Author  nkirby (11104) 
Entry type  Definition 
Classification  msc 51M25 
Related topic  SurfaceOfRevolution2 