Question

A rectangle with its base on the x axis is to be inscribed under the positive portions of the graph of y=8-x^2 . Write an equation for the area of the rectangle which uses only the variable x.

Answer 1

y is positive from x = -sqrt8 to +sqrt8

The area is Integral (8-x^2) dx

from x = -sqrt8 to +sqrt8

Answer 2

eh? We're not looking for the area under the parabola. We want the area of a rectangle inscribed under it.

Let the width of the rectangle be 2x. That is, x is the distance from (0,0) to a bottom corner of the rectangle.

the area is thus
f(x) = 2xy = 2x(8-x^2)

The domain of f is obviously [-√8,√8]