Calculus homework help

Question

A solid has a base bounded by y= 9- x² and the x-axis. Each cross-section perpendicular to the base in parallel to the x-axis is a rectangle of height 4. Find its volume.

Answer 1

Since y = 9 - x² is symmetrical about x = 0 (the y-axis), then the width of each rectangle is twice the "x-coordinate".

Since y = 9 - x² then x² = 9 - y or x = √(9 - y) making the width of each rectangle 2√(9 - y).

So, since the area of each rectangle is width times height, then the area of each cross sectional rectangle is (4)(2√(9 - y) or 8√(9 - y). The volume of one of these would be the cross sectional areaa times its thickness or (8√(9 - y))(dy) so now just integrate between y = 0 to y = 9:

V = ₀∫⁹ 8√(9 - y) dy

You can find the antiderivative using the power rule or you can plug this in a calculator.