If you know how to evaluate double integrals it's easy.

The area between f(x) and g(x) is

or

a and b are x coordinates, c and d are y coordinates of the intersection points of the curves.

Example:

Find the area of the circle .

Solution(analytic, not geometric ;p)

This gives us two functions of x:

So we'll find the area between these curves.

Find the x coordinates of the intersection points:

So the integral is,