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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 = [tex]\int divF dV[/tex]

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

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!

Equations: divF = dF1/dx + dF2/dy + dF3/dz

Flux = [tex]\int divF dV[/tex]

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

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!

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