Stokes Theorem Problem: Surface Integral on Ellipse with Curl and Normal Vector

[astonmartin]

Homework Statement

F = xi + x³y²j + zk; C the boundary of the semi-ellispoid z = (4 - 4x² - y²)^1/2 in the plane z = 0

Homework Equations

(don't know how to write integrals on here, sorry)

double integral (curl F) . n ds

The Attempt at a Solution

curl F = 3y²x²k
n = k

curl F . n = 3y²x²

So I have a surface integral, which I think I can change to dA since the differential of the surface area is just 1dA...

Now this is where I'm stuck. How do i do the double integral with an ellipse? I tried it in rectangular coordinates but got some function I don't know how to integrate. Help!

 

[Billy Bob]

Can you do this by integrating F dot dr over the boundary curve instead?

r(theta)=cos(theta) I+2sin(theta) J+0 K

 

[Dick]

What function do you get that you can't integrate? It looks like it's just a trig substitution to me.

 

[cartonn30gel]

what you can do is use Stoke's theorem to convert to the line integral of the vector filed and then if you use Green's theorem to convert that to a double integral, use a trig sub and then integrate. (integration takes a little bit of work) I think the answer is pi. Anybody agree?

 

[Dick]

Yes, it's pi. Both ways.