Double Integrals and circles - Confirmation Wanted

[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

 

[TD]

Have you covered polar coordinates?

 

[Nima]

Have you covered polar coordinates?

Hi, no unfortunately I haven't covered polar co-ordinates yet.

mmm so yes I see that f(x, y) = r and now we have 2 circles with radii r = 4 and r = 1 respectively.

Could you explain to me how to do this Q if that's ok? Thanks.

 

[DMinishedCapacity]

Just think about it. from what pts are we integrating wrt the radius? Then, what angle to what angle are we integrating (wrt theta). drawing a picture is helpful.