Integration/Double Integrals Advice Required

[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 separate my variables...

Thanks

 

[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.

 

[TD]

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

 

[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)

 

[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.

 

[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.