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tulip007

Member

Registered: 2008-04-30

Posts: 10

elliples and double integrals

Express the area A defined by the upper quadrant of the ellipse x^2/a^2  + y^2/b^2 = 1
as a double integral in the x,y plane.  Hence show that A = πab/4

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: elliples and double integrals

In the upper-right quadrant,

There are two ways of writing the double integral. You can integrate with respect to y from 0 to y and then with respect to x from 0 to a, which would be

Or you can integrate with respect to x from 0 to x and then with respect to y from 0 to b.

In the first case, you get

and you can evaluate the integral using the substitution

. Same for the second case.

Last edited by JaneFairfax (2008-05-14 00:28:04)