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Solids of Revolution

Posted 10 years ago
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I know it's possible to do plots of revolution in Mathematica, but is it possible to also visualize them in WolframAlpha?  For example, I have a problem that says the following:

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.
x = y - y^2, x=0; about the y-axis

Entering this didn't give me the results I expected. Is this possible?
POSTED BY: Ryan Hefner
The area below the curve above x = 0 is
In[4]:= Integrate[y - y^2, {y, 0, 1}]

Out[4]= 1/6
The look of the surface of revolution is given by
RevolutionPlot3D[y - y^2, {y, 0, 1}]
The get the volume remember what one does to get the volume of the torus: If R is the radius of rotation of the circle with radius r (the actual tube of the torus), then the volume of the torus is 2 Pi R Pi r^2 = 2 Pi^2 R r^2, R >= r. Rotating your curve, obviously R = 1/2 so the volume is 2 Pi 1/2 1/6 = Pi/6. Check this with WolframAlpha

querying "Volumn of revolution" and using the given formula
In[6]:= 2 \[Pi] Integrate[y (y - y^2), {y, 0, 1}]

Out[6]= \[Pi]/6
In Wolfram Alpha one can not do that with exactly one query seemingly, but after one typed "volume of revolution" one sees the formula from In[6] above and then on might put it into WolframAlpha to get the integral there too, so one does it with two queries. 

By the way, nearly every command from Mathematica might by typed into and answered by WolframAlpha, the restrictions are only
  • the number of characters of the command
  • the computational cost to get the answer
If you need more from WolframAlpha, consider to subscribe it as a "Pro" User.
POSTED BY: Udo Krause
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