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Integral of function over ellipse

by nickthequick
Tags: ellipse, function, integral
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nickthequick
#1
Feb27-13, 01:27 PM
P: 52
Hi,

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

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

Any suggestions are appreciated!

Thanks,

Nick
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Whovian
#2
Feb27-13, 02:35 PM
P: 642
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
#3
Feb27-13, 02:38 PM
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LCKurtz's Avatar
P: 7,586
Try the substitution$$
\frac x a = r\cos\theta,\,\frac y b = r\sin\theta$$

nickthequick
#4
Feb27-13, 06:10 PM
P: 52
Integral of function over ellipse

Got it!

Thanks


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