First you write the equation for the curve as a function of x: f(x) = equation.
Area = the integral [from 0 to max x] f(x)dx. Turning the crank gives the answer.
Surround the curve with a box. Generate random points that will land inside the box. Determine whether each generated point is inside the curve or outside. If 62% of the random points lie within the curve, the area of the curve is 62% of the area of the box. Obviously precision grows as a function of the sheer number of points.