Double integral with limit with Monte Carlo simulation

I'm trying to find the solution to the integral

over the region between two circles x^2+y^2=1 and x^2+y^2=4. I used a few different methods before giving up. The limits are giving me a hard time. Can anybody help me? Thanks in advance.

Answer 1

[Torsten]

Hint: Use polar coordinates.

Best wishes

Torsten.

Answer 2

[Abhigyan Malviya]

You can solve that easily using Monte carlo sampling. To begin, your limits cover the area between two circles. As such, you need to sample the points whose x,y coordinates are inside those circle. Further, the function is monotonically decreasing over the limit, as such the maxima of the curve can be obtained easily.

Next choose random points over the volume using Monte Carlo scheme, finally take the ratio of the volumes (between the cylinder formed by the two circles and Z=maxima) and the points inside the curve. In case you have no clue of Monte Carlo integration scheme, have a look here.

You can contact me if you need more help: www.lennardjones.com

Regards,

Abhigyan Malviya