Question

Triple integrals

Find the volume of the wedge cut from the cylinder 4x^2+y^2=9 by the planes z=0 and z=y+3

How would you solve this?

Answer 1

Another triple integral question

Let G denote the solid region in the first octant inside the cylinder x^2+(y-4)^2=16, and bounded above by the sphere x^2+y^2+z^2=81. Evaluate the triple integral of (z^3)/(81-x^2-y^2)^2

Answer 2

Start by drawing a picture of the space we are interested in. That's what I'm doing now.

Answer 3

well the problem I'm having right now with the first one is that I'm not even sure what variables to be integrating in. On one hand, I could try polar, but since there's a constant 4 on the x^2, I would have to create a jacobian and probably substitute 2rsintheta in for y. I'm not sure how that would work in comparison to having x=u and y/2=v though. Any ideas?

Answer 4

I was thinking cylindrical coordinates; however, I'm still messing around with it.

Answer 5

Do you have an answer for the first question?

Answer 6

I would say the answer to the first question is 81/4 * pi. It has been a while since I have done these, so this is a good exercise for me. Since we are using a triple integral to find the volume of the given space I wrote that
\[Volume =\int\limits_{0}^{2\pi}\int\limits_{3/2}^{3}\int\limits_{0}^{rsin \theta + 3}r dz dr d \theta \]



That's my thinking behind my answer.