1. ## Finding Flux

Another question I'm having trouble with is:

"Find the flux of F(x,y,z) = -yi + xj + (6z^2)k out of the closed surface S, bounded by the paraboloids z = 4 - x^2 - y^2 and z = x^2 + y^2."

I can get the graph drawn easily enough, I just can't figure out what the limits of my integration should be.

Any assistance would be much appreciated.

2. Originally Posted by Katzenjammer
Another question I'm having trouble with is:

"Find the flux of F(x,y,z) = -yi + xj + (6z^2)k out of the closed surface S, bounded by the paraboloids z = 4 - x^2 - y^2 and z = x^2 + y^2."

I can get the graph drawn easily enough, I just can't figure out what the limits of my integration should be.

Any assistance would be much appreciated.

the domain is $\displaystyle D = [(x,y,z)\in \mathbb{R}^3 | x^2 + y^2 \leq z \leq 4 - x^2 - y^2 , x^2 + y^2 \leq 2 ]$

or cylinder coordinates :

$\displaystyle D = [(r,\theta,z) \in \mathbb{R}^3 | r^2 \leq z \leq 4 - r^2 ~, 0 \leq r \leq \sqrt{2} ~, 0 \leq \theta \leq 2\pi ]$

3. This may sound dumb, but can I break this into two separate flux integrals? Since they are two paraboloids that meet at z=2, is that a good way to go about it?

And I'm thinking the top paraboloid would be "oriented upward" and the bottom paraboloid would be "oriented downward", yes?

4. you have to break it into 2 parts

if the orientation is outward

the upper part has formula z= 4 - x^2 -y^2

with Normal N = 2x i +2y j + k

For the bottom z = x^2 + y^2

With N = 2x i +2 y j - k

The domain of integration however is the same in both cases