Where is called the jacobian determinant and it's given by:
Now, looking at the region in rectilinear coordinates can give you some hints as to the limits in polar coordinates.
Now, the 2nd constraint on the region says that . You should be able to see that this equation describes a circle whose radius is less than , in other words, the radius is limited by .
Now, the 1st constraint on the region says that . Now, we know that the line at which y= x is a diagonal line going through the origin. And y is greater than x in any point that lies ABOVE that line. In other words, the region in question is made up of points above or on the line y = x. This means that your theta is limited by .
You can see all of this visually by drawing it out. Go on, do this:
1) Draw the x-y coordinates.
2) Draw a circle centred at the origin with a radius of .
3) Draw the line y = x.
4) Now shade the region that lies above the line y = x, but does not go outside the circle.
You should see that this region is bounded by:
Hence, in this particular situation: