solid of revolution

Let y=f(x) be a curve for x in an intervalMathworldPlanetmathPlanetmath [a,b] satisfying f(x)>0 for x in (a,b). We may construct a corresponding solid of revolutionMathworldPlanetmath, say 𝒱={(x,y,z):x[a,b] and y2+z2f(x)2}. Intuitively, it is the solid generated by rotating the surface yf(x) about the x-axis.

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

  • the interior of a cylinderMathworldPlanetmath of radius c>0 and height h with f(x)=c for 0xh,

  • the interior of a sphere of radius R>0 with f(x)=R2-x2 for -RxR, and

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

Let Γ be a simple closed curve with parametrization α(t)=(X(t),Y(t)) for t in an interval [a,b] satisfying Y(t)0 for t in [a,b]. By the Jordan curve theoremMathworldPlanetmath, we may choose the set of points, 𝒮, ”inside” the curve, i.e. let 𝒮 be the boundedPlanetmathPlanetmathPlanetmath connected componentMathworldPlanetmathPlanetmathPlanetmath of the two connected components found in 2Γ. Another sort of solid of revolution is given by 𝒱={(x,y,z):x=X(t) for some t in [a,b] and y2+z2=s2 for some s such that (x,s)𝒮Γ}. Intuitively, it is the solid generated by rotating 𝒮Γ 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 α(t)=(rcost,rsint+R) for 0t2π,

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

    α(t)={(Rcosπt,Rsinπt) if t[0,1](r(1-t)+R(t-2),0) if t[1,2](-rcosπt,rsinπt) if t[2,3](r(4-t)+R(t-3),0) if t[3,4].
Title solid of revolution
Canonical name SolidOfRevolution
Date of creation 2013-03-22 17:19:57
Last modified on 2013-03-22 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