Register to reply

Integral of function over ellipse

by nickthequick
Tags: ellipse, function, integral
Share this thread:
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
Phys.Org News Partner Science news on Phys.org
Physical constant is constant even in strong gravitational fields
Montreal VR headset team turns to crowdfunding for Totem
Researchers study vital 'on/off switches' that control when bacteria turn deadly
Whovian
#2
Feb27-13, 02:35 PM
P: 643
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
HW Helper
Thanks
PF Gold
LCKurtz's Avatar
P: 7,721
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


Register to reply

Related Discussions
Set up polar area integral of ellipse Calculus & Beyond Homework 3
Green's theorem- integral over an ellipse Calculus & Beyond Homework 1
Integral over a Rotated Ellipse Calculus & Beyond Homework 2
Line integral around an ellipse Calculus & Beyond Homework 8
Area of ellipse integral. Calculus & Beyond Homework 13