Evaluate the given integral by changing to polar coordinates.
Thanks in advance.
The integral is . You need to work out the region over which the integral takes place, and for that you must draw a picture.
Notice that y goes from 0 to . The upper limit of that interval occurs when . Square both sides of that equation, rearrange it a bit, and you see that it can be written as . You will recognise that as the equation of the circle centred at (1,0) with radius 1. But the lower limit for y is 0, so the region of integration is the upper half of that circle.
That is stage 1 of the solution. Stage 2 is to describe that region in terms of polar coordinates. Have you drawn a picture of that semicircular region yet? If so, you should be able to see that goes from 0 to and, for each fixed value of , r goes from 0 to .
Once you have got that far, you can write the integral as , and the rest should be easy.