How to 3D plot a volume of revolution?

Questions
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1

Hi

I’m currently teaching how to calculate volumes of revolution using integration,
for e.g. suppose f(x)=x^2 is rotated 360 degrees about the x-axis from x = 2 to x=4, the volume generated would be given by

Integral[Pi(f(x))^2,2,4]

What I’m trying to do is plot the generated volume for explanation and demonstration purposes.

some of the more complicated examples are something like,
The region bounded by the curve y=9-x^2, x=2 and the x-axis is rotated 2Pi radians about the y-axis, calculate the volume of the solid generated.
I know how to calculate the actual volume, but is there any easy way of plotting the solid formed in mathematica.

2

Dear Hifas,

are you looking for something like this?

RegionPlot3D[
 Sqrt[y^2 + z^2] < x^2, {z, -16, 16}, {y, -16, 16}, {x, 2, 4}]

which gives

Rotation.jpg
Rotation.jpg388×416 32.4 KB

3

RevolutionPlot3D is also very useful for plotting rotated functions in general if you want to show it done with less math.

4

Dear Sean,

I thought that RevolutionPlot3D only shows the surface, i.e.

RevolutionPlot3D[Sqrt[t], {t, 0, 4}, AspectRatio -> 1]

revolution_m.jpg
revolution_m.jpg1062×1178 67.1 KB

I thought that the question was how to plot a “solid”, i.e the volume not the surface. I tried the option “Filling”, but that does apparently not work. Is there an option to get RevolutionPlot3D to draw solids?

BTW, there is this nice demonstration:

[http://demonstrations.wolfram.com/SolidsOfRevolution/

](http://demonstrations.wolfram.com/SolidsOfRevolution/)but as far as I can see the author did not use RevolutionPlot3D either.