Write an expression for the rectangle's height. For a parabola that opens upward, the two lower corners of the rectangle lie on the curve of the parabola, and the two upper corners lie on the x-axis. Define the lower right hand corner as (x, f(x)), and the other corner points would be (-x, f(x)), (-x, 0) and (x, 0), where f(x) = y. Therefore the the height is h = -f(x). For example, if the rectangle is bound by the parabola f(x) = x^2/2 - 8, the height is h = 8 - x^2/2.
Write an expression for the rectangle's width. Since the two upper points are (-x, 0) and (x, 0), the width is w = 2x.
Write the equation for the area. For example, since the area is the width times the height, A = 2x(8 - x^2/2), which simplifies to A = 16x - x^3.
Take the derivative of the equation with respect to x. For example, dA/dx = 16 - 3x^2.
Set the derivative equal to zero, and solve for x. By setting the derivative to zero, you are specifying only those values of x where there is a local maxima or minima. For example:
16 - 3x^2 = 0
3x^2 = 16
x = sqrt(16/3)
x = 2.31
Plug the value of x into the equation to find the area of the rectangle. For example:
A = 16x - x^3
A = 16*2.31 - (2.31)^3
A = 24.6