1. Jan 30, 2006

Nima

Hey, my Q is:

"Integrate f(x, y) = Sqrt(x^2 + y^2) over the region in the x-y plane bounded by the circles r = 1 and r = 4 in the upper half-plane".

Well, I firstly sketched out the region I get as my area in the x-y plane. I deduced that the ranges for x and y are:

0 <= x <= 4
Sqrt[1 - x^2] <= y <= Sqrt[16 - x^2]

1.) Is this right?
2.) How do I then calculate the integral of f(x, y) over this region? I know I'm doing a double integral but I don't see how I can seperate my variables...

Thanks

2. Jan 30, 2006

Moo Of Doom

That integral is screaming polar coordinates! f(x,y)=r and the region is circular. It'll be really easy in polar coordinates.

3. Jan 30, 2006

TD

There is no need to post every problem twice...

4. Jan 31, 2006

TheDestroyer

Use the polar coordinates where ds=r dr d(theta) or if you want to use rectangular use these limits

x 1 -> 4
y sqrt(1-x^2) -> sqrt(16-x^2)

5. Jan 31, 2006

Moo Of Doom

If he wanted to use rectangular, he'd need to split it into 3 parts because when |x|>1, there is no inner circle anymore. So, as x goes from -4 => 1, 0 < y < sqrt(16-x^2), as x goes from -1 => 1, sqrt(1-x^2) < y < sqrt(16-x^2), and as x goes from 1 => 4, 0 < y < sqrt(16-x^2) again. Or he could do the whole half-disk of radius 4 and subtract the half-disc of radius one from it.

6. Feb 1, 2006

TheDestroyer

yes you are right, but i just ment that he could understand that in this example we can multiply my previous result with 2 to get the answer.