Help with a triple Integration Question?

Hi I am having difficulty with the following question

2. Using cylindrical polar co-ordinates r, theta , z, evaluate

∫∫∫B z^7 dV; where B is the finite body between the plane z = 0 and the paraboloidal surface S : z = 36 − x^2 − y^2.

I have managed to find the limit of theta which is 2pi and 0, but I am struggling to find it for r and z. For z I think one of the limits is 36-r^2 and the other limit for z is 0 (do you think this is right), also what would the limits of r be ( would it be 6 and -6 or 6 and 0)

Re: Help with a triple Integration Question?

In cylindrical coordinates, which is what what the problem specifically asks you to use, the paraboloidal surface is $\displaystyle z= 36- r^2$. It should be easy to see that z is never more than 36. And you are told that z is not below 0. **For each z**, r goes from 0 to $\displaystyle r= \sqrt{36}= 6$. Or you could argue that r goes from 0 to a largest value of 6 (where z= 0) and, **for each r**, z goes from 0 to $\displaystyle 36- r^2$.

Re: Help with a triple Integration Question?

Quote:

Originally Posted by

**HallsofIvy** In cylindrical coordinates, which is what what the problem specifically asks you to use, the paraboloidal surface is $\displaystyle z= 36- r^2$. It should be easy to see that z is never more than 36. And you are told that z is not below 0. **For each z**, r goes from 0 to $\displaystyle r= \sqrt{36}= 6$. Or you could argue that r goes from 0 to a largest value of 6 (where z= 0) and, **for each r**, z goes from 0 to $\displaystyle 36- r^2$.

Thanks :) - for the theta limits I thought it was 2pi to 0 but now I think it might be pi/2 to 0, what do you think?