# Finding the flux (Divergence Theorem)

## Homework Statement

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

## Homework Equations

Divergence theorem equation stated in the attempt part

## The Attempt at a Solution

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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.

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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.
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?

CompuChip