First comment is about the statement of the question. I don't know if this is just a cultural difference between Canada and Britain, but when I say "integral from a to b", I mean that a is the lower limit and b is the upper limit. These question all seem to be phrased the other way round, so that Integral(from 1 to (1/sqroot(2))) must presumably mean that 1/sqroot(2) is the lower limit and 1 is the upper limit.

So it looks as though the first integral should be .

The hardest part of the question is probably to figure out the region over which you are supposed to be integrating. In this case, y goes from the circle to the line , and x goes from (which is where the line crosses the circle) to x=1.Draw a pictureto see that the region of integration is the bounded by the unit circle, the line y=x and the line x=1. You will then be able to see from the picture that this region is described in terms of polar coordinates as follows: θ goes from 0 to π/4, and for each fixed value of θ, r goes from 1 to sec(θ). So the integral becomes . Once you have got that far, the actual evaluation of the integral should be easy.

Do the other questions the same way. The key step is todraw a pictureof the region of integration.