Thread: Find the volume inside a cone and a cylinder using change of variables

I have been trying to figure this question out for a while with no luck. I have to find volume inside the cone z=2a−sqrt(x^2+y^2)and inside the cylinder x^2+y^2=2ay. I am trying to use cylindrical coordinates but I am just having some issues. If anyone could help I would appreciate it.

What are the issues? Are they with

$\displaystyle \int_0^{2\pi}\ \int_0^{a \sin \theta}\ \int_0^{2a - r}\ r\ dz\ dr\ d\theta$

or thereafter?

Originally Posted by tom@ballooncalculus

What are the issues? Are they with

$\displaystyle \int_0^{2\pi}\ \int_0^{a \sin \theta}\ \int_0^{2a - r}\ r\ dz\ dr\ d\theta$

or thereafter?

Yes my issues are with the above statement. I understand how you got boundaries for the integral however when I did it I got an answer of a^3*pi which is not the answer in the back of the book. The answer in the back of the book is a^3(2pi-32/9)... Now I don't know if I computed the integral incorrectly or what but if you could confirm the calculation it would be a huge help.

Typo, we meant...

$\displaystyle \int_0^{2\pi}\ \int_0^{2a \sin \theta}\ \int_0^{2a - r}\ r\ dz\ dr\ d\theta$

... yes? But I agree with your calculation (now multiplyied by 4). So there may be a difference of geometric interpretation? Hmm...