Question

1a) suppose we want to integrate a funciton f(x,y) over the ellipse x^2/a^2 + y^2/b^2=1 (where a is not equal to b are positive constants). Suggest a suitable change of variables, and find the Jacobian of this transformation. (use a modified version of polar coordinates)

b) Using the answer to (a), find the volume of the region enclosed beneath the elliptic paraboloid z = 12-x^2-3y^2 and above the xy-plane.

Answer 1

I have no idea where to start. I tried to convert the coordinates into polar coordinates (x=rcos(theta), y=rsin(theta). I don't know even where to start...

Answer 2

A much better parameterization would be x = a cos (theta), y = b* sin(theta)

Answer 3

okay so, I have donethat and got the determinant of the jacobian =abr

how do i do part 2?....

Answer 4

There should be no r involved, I'm not sure what you did there.

Answer 5

Okay, I read this a little more closely and my thought is this: The first question is, to me, poorly posed because your domain of integration is a 1-dimensional curve. You only need one coordinate to specify where you are on that ellipse (theta).

If you're trying to integrate over the domain enclosed WITHIN the ellipse, the situation changes. Now you need to consider your variable r. I would suggest a transformation similar to what you had before, but rather
x = (r/a) cos(theta), y = (r/b) sin(theta)

Answer 6

I'm sorry, that should be (r*a) and (r*b), not (r/a) and (r/b)

Answer 7

I think we figured it out...I'm not sure though...the assignment's not due till the end of the week, haha

Answer 8

THANKS!

Answer 9

Rather than depending on the Internet to get help on the assignment, you can instead visit me during my office hours.