We have a cylinder of radius a and height h. We need to prove its volume to be equal to pi*a^2*h using triple integral and spherical coordinates. The best way to solve this problem is to divide the cylinder to two volume regions , the first region is the one defined by phi to range from 0 to arctan(a/h) and the second region is the one defined by phi to range from arctan(a/h) till pi/2. Surley , for both regions theta will vary from 0 till 2*pi. However , i tried many times to find the proper limits for the radius for each region but i failed. So how do i properly define the radius for each region ?

Edit : I know the limits of each region for the radius r= 0 till asecθ and r= 0 till bcscθ but my question exactly is how to drive them ?

  • $\begingroup$ Please see math.meta.stackexchange.com/questions/5020 $\endgroup$ – Lord Shark the Unknown Nov 13 '18 at 21:03
  • $\begingroup$ The equations for vertical and horizontal lines in polar coordinate are $r=a \sec \theta$ and $r=b\csc \theta.$ You can adapt these readily. $\endgroup$ – B. Goddard Nov 13 '18 at 21:07
  • $\begingroup$ @LordSharktheUnknown Ok $\endgroup$ – John adams Nov 13 '18 at 21:08
  • $\begingroup$ @B.Goddard Please explain how do you drive them $\endgroup$ – John adams Nov 13 '18 at 21:08
  • $\begingroup$ Start with $y=a$ is the same as $r\sin \theta = a.$ $\endgroup$ – B. Goddard Nov 13 '18 at 21:27

I would say that to find the volume of a cylinder use cylindrical coordinates.

But you want to use spherical.

Your boundaries

$x^2 + y^2 = a^2\\ z = 0\\ z = h$

$x = \rho\cos\theta\sin\phi\\ y = \rho\sin\theta\sin\phi\\ z = \rho\cos \phi$

$\rho^2\cos^2 \theta\sin^2\phi + \rho^2\sin^2 \theta\sin^2\phi = a^2\\ \rho^2\sin^2\phi = a^2\\ \rho = a\csc\phi$

$\rho\cos \phi = h\\ \rho = h\sec \phi$

$a\csc\phi = h\sec \phi\\ \tan\phi = \frac ah$

$\int_\limits0^{2\pi}\int_\limits0^{\arctan \frac {a}{h}}\int_\limits0^{h\sec \phi} \rho^2\sin \phi \ d\rho\ d\phi\ d\theta\\ + \int_\limits0^{2\pi}\int_\limits{\arctan \frac {a}{h}}^\frac {\pi}{2}\int_\limits0^{a\csc \phi} \rho^2\sin \phi \ d\rho\ d\phi\ d\theta$

  • $\begingroup$ How did you drive the limits of upper limits of the radius for each region ? $\endgroup$ – John adams Nov 13 '18 at 21:17
  • $\begingroup$ @Johnadams Did you ask this question before I completed my last edit? Is it all clear now? $\endgroup$ – Doug M Nov 13 '18 at 21:29
  • $\begingroup$ Yes , Thank you very much. $\endgroup$ – John adams Nov 13 '18 at 21:30

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