How Do I Set Bounds for These Integrals Correctly?

[dylanhouse]

Homework Statement

I am having trouble setting up the bounds on the following two integrals:

(a) The region E bounded by the paraboloid y=x²+z² and the plane y=4.
(b) The region bounded by the cylinder x²+y²=1, z=4, and the paraboloid z=1-x²-y².

Homework Equations

The Attempt at a Solution

I thought for (a) to use 4 < y < x²+z²
-sqrt(y-z²) < x < sqrt(y-z²)
-sqrt(y-x²) < z < sqrt(y-x²)
But these don't seem right.
I'm not sure where to begin for (b) except that z may be upper bounded by 4?
Thanks in advance.

 

[Orodruin]

(a) The conditions you put on y will not give you a bounded region. For simplicity, I suggest you start working with $r=\sqrt{x^2 + z^2}$ instead of x and z.

(b) As in (a), polar coordinates will serve you well here.

 

[pasmith]

For these you want to use cylindrical coordinates. For (a), you should take $(x,y,z) = (r \cos \theta, y, r \sin \theta)$. For (b), you should take $(x,y,z) = (r \cos\theta, r \sin \theta, z)$.