Is it possible to calculate the following with a triple integral, and if so, how?
The volume of a solid bounded by two cylinders: radius = 1 and radius = 2, AND by two cones: $\varphi$ = $\pi$/6 and $\varphi$ = $\pi$/3.
I don't believe it's possible using only spherical coordinates or only cylindrical coordinates, for the following reason: (it wouldn't allow me to post the image directly as this is my first post here).
When you go to attempt to pick a range for $\varphi$ you will always have some unaccounted for volume as is represented by the (poorly drawn) image above.
Could you complete the integral by making a change in variables from spherical to cylindrical halfway through the formula? Or would you be stuck computing two separate triple integrals (the second being to account for the volume represented in the image above)?
Any help or insight would be greatly appreciated.