Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = x^2, x = y^2; about y = 1

Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

y = x²,  x = y²;    about y = 1

Answer 1

First, find the area between the 2 curves which is just ∫(x^(1/2) - x^∫2) dx from x = 0 to 1 which is 1/3 = Area

Then find the y coordinate of the center of area (y-bar) which is { ∫∫ y dy dx }/Area where y varies from x² to x^(1/2) and x varies from 0 to 1. This is the center of area for the area between the 2 curves.

So, y-bar = (3/20)/(1/3) = 9/20.

The distance from the line y = 1 and 9/20 is 11/20 = R, (pi = 3.14159..)

Then using the Pappus Theorem, the volume of revolution is V = (2pi)(R)(Area) = 11pi/30