calculating volume

Question

Let R be the region in the cycle coordinate plane above y=x and bounded below by y= x^2.

calculate the volume of f the solid S obtained by revolving R about the y axis

Answer 1

Using the shell method:

The curves intersect when x = 0 and x = 1

A typical vertical cross section at an average distance of x units from the y-axis has width dx and

height x - x².

Volume of shell = 2π(radius)(height)(thickness) = 2π(x)(x - x²)dx = 2π(x² - x³)dx

Volume of solid = 2π∫ _{(from 0 to 1)} [ x² - x³ ] dx Evaluate!