First step, draw a picture. x lies between 0 and 1/2 and, for each x, y is between the straight line and the unit circle .

In polar coordinates, the unit circle is given, of course, by r= 1 and, since the slope of a line is the tangent of the angle the line makes with the x-axis, the line makes an angle [tex]\pi/3[tex]. Further, that line just happens to cross the circle where : x= 1/2.

The region in this integral s the segment of the unit circle in which r goest from 0 to 1 and goes from to .