Integral of function over ellipse

[nickthequick]

Hi,

I'm trying to find
\iint_S \sqrt{1-\left(\frac{x}{a}\right)^2 -\left(\frac{y}{b}\right)^2} dS

where S is the surface of an ellipse with boundary given by \left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2 = 1.

Any suggestions are appreciated!

Thanks,

Nick

 

[Whovian]

Do you mean the interior of an ellipse?

Anyway, the first thing I though of is Green's Theorem for some reason. Probably since then we can make the substitution $\left(\dfrac xa\right)^2+\left(\dfrac yb\right)^2=1$.

The second thing I thought of was a change of coordinates and a multiplication by the Jacobian determinant, then we have it reduced to

$$a\cdot b\cdot\iint_C\sqrt{1-m^2-n^2}\mathrm{d}S'$$

where C is the unit circle wrt m and n and S' should have a fairly obvious definition.

 

[LCKurtz]

Try the substitution$$
\frac x a = r\cos\theta,\,\frac y b = r\sin\theta$$

 

[nickthequick]

Got it!

Thanks