Help doing an integral using stokes theorem?

[madcattle]

Homework Statement

F= xi + x³y²j + zk
C is the boundary of the semi-ellipsoid z=√(4-4x²-y²) in the plane z=0

Homework Equations

Stokes theorem states:
∫∫(curlF ° n)dS

The Attempt at a Solution

I found the curl of the F to be 3x²y²k
I found that the dot product of CurlF and n = 3x²y² divided by dS

Then,
∫∫(3x²y²)/(dS)*dS
=∫∫(3x²y²)dxdy
I evaluated this integral on Wolfram Alpha with the following boundaries:
y goes from -√(4-4x²) to √(4-4x²)
and x from -1 to 1 and got the correct answer = ∏

However, I am finding it impossible (for me!) to do the integral by hand and am wondering if someone can help me turn this into polar coordinates or something else that makes it more solvable

Thank!

 

[Citan Uzuki]

You want to integrate on the region $x^2 + \left( \frac{y}{2} \right)^2 \leq 1$. Consider the substitution $y = 2y_1$. Then your integral becomes:

$$\int_{D} 24 x^2 y_1^2\ dx dy_1$$

Where D is now the unit disk in the (x, y_1) plane centered at the origin. Can you see how to put that into polar coordinates?