Flux through a paraboloid? The Divergence Theorem and Integration Error

[Raen]

The Problem: I have a paraboloid open along the positive z-axis, starting at the origin and ending at z = 100. At z=100, the horizontal surface is a circle with a radius of 20. Water is flowing through the paraboloid with the velocity F = 2xzi - (1100 + xe^-x^2)j + z(1100 - z)k. I'm asked to find the flux through the paraboloid using the divergence theorem.


Equations: divF = dF1/dx + dF2/dy + dF3/dz
Flux = $$\int divF dV$$



My attempt: I started by finding the divergence.

2z + 0 + (1100 - 2z) = 1100

Next, I found the equation for the paraboloid.

z = r^2/x
100 = 20^2/x
100 = 400/4
z = r^2/4

Then I iterated the integral.

Flux = $$\int^{2\pi}_{0} \int^{20}_{0} \int^{100}_{r^2/4} 1100r dz dr d\theta$$

When I solve the integral, I end up with -11000000pi, but the answer is supposed to be 10000pi. Where am I going wrong?

If I have it correct thus far and my problem is in my integration, please let me know and I'll type out the integration step by step, as well. Thank you!

 

[tiny-tim]

Hi Raen!

(try using the X² icon just above the Reply box )

The Problem: I have a paraboloid open along the positive z-axis, …

(i haven't checked your figures, but …)

The paraboloid is open, so you'll need to subtract the flux through the open end.

 

[Raen]

The flux through the top is positive, so subtracting the flux through the open circle from what I already had gives me an even larger negative number, -51000000pi.

 

[tiny-tim]

oooh, I didn't notice that minus sign!

but how did you manage to get a negative result from integrating something that's everywhere positive?