Finding the flux (Divergence Theorem)

1. Apr 17, 2009

nb89

1. The problem statement, all variables and given/known data
By using divergence theorem find the flux of vector F out of the surface of the paraboloid z = x^2 + y^2, z<=9, when F = (y^3)i + (x^3)j + (3z^2)k

2. Relevant equations
Divergence theorem equation stated in the attempt part

3. The attempt at a solution

2. Apr 17, 2009

CompuChip

Suppose you are doing the z-integral first, so we are looking at a slice of fixed z. Then the slice looks like a disk of radius rmax. Instead of 0 to 3, you want to integrate each slice over r from 0 to rmax. You can express rmax in terms of z.
So you will get
$$\int_0^9 \int_0^{r_\mathrm{max}(z)} \int_0^{2\pi} 6 z r \, \mathrm{d}\varphi \, \mathrm{d}r \, \mathrm d{z}$$
where rmax(z) depends on z instead of being identically equal to 3 as you have now.

Last edited: Apr 17, 2009
3. Apr 17, 2009

nb89

I understand what you mean about the radius changing with z, but how would i integrate dr with limits rmax(z)?
Also what about the Fn ds double integral?

4. Apr 18, 2009

CompuChip

For some given z, what is the maximum radius (i.e. the upper boundary of your r-integration)?

I haven't looked into the other approach (where you first apply the divergence theorem) but it does look a bit more complicated to me (finding the correct normal unit vector and all).

5. May 2, 2009

nb89

Just thought id bump this question. I understand my error when calculating the volume integral, the problem I'm left with is the surface integral. I still cant spot the mistake there. Any suggestions? Thanks.