Find the volume of the solid whose base is the semicircle y= squareroot( 16-x^2), where -4< x< 4 and whose cross section perpendicular axis are squares

Answer 1

The volume of a typical cross section is y^2dx = (sqrt[16-x^2])^2 dx

= (16 - x^2)dx

By the symmetry of the solid, we can find the volume by first finding the volume of the portion of the solid where 0<x<4 and then double it.

Volume = 2[integral (from 0 to 4) of (16 - x^2)dx]

= 2(16x - (1/3)x^3) evaluated from 0 to 4

= 2(64 - 64/3) = 256/3