I think the correct formula is .
To do the integral, you'll need the identity .
To check your answer, the result should be roughly the area of a circle with radius 2, which, of course, is .
If you look at a small piece of the graph, it is a sector of a circle (the shape of a slice of pie). The angle at the origin is , and the radius is . Since the angle at the origin for a whole circle is , our wedge shape is of a circle. Since the area of a circle is , the area of the wedge is . That explains the integrand.
Our curve is traced by going from 0 to , so that explains the limits of integration.