# Integral of function over ellipse

by nickthequick
Tags: ellipse, function, integral
 P: 48 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
 P: 634 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.
 PF Patron HW Helper Thanks P: 6,756 Try the substitution$$\frac x a = r\cos\theta,\,\frac y b = r\sin\theta$$
P: 48

## Integral of function over ellipse

Got it!

Thanks

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