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Integral of function over ellipse 
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#1
Feb2713, 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 


#2
Feb2713, 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 $$a\cdot b\cdot\iint_C\sqrt{1m^2n^2}\mathrm{d}S'$$ where C is the unit circle wrt m and n and S' should have a fairly obvious definition. 


#3
Feb2713, 02:38 PM

HW Helper
Thanks
PF Gold
P: 7,586

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


#4
Feb2713, 06:10 PM

P: 52

Integral of function over ellipse
Got it!
Thanks 


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