Use the method of cylindrical shells to find the volume of the solid obtained by rotating the bounded region..

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis:

y= x^2 , y = 2 - x^2 ; rotated about x = 1

Any and all help is much appreciated! Thanks so much in advance!

Hello


Both curves intersect at x = -1 and +1. So you should integrate from -1 to +1. But if you shift the curves to the right by one unit, you can rotate around the y axis. And this is then identical to rotating the original curves around x = 1, because both original curves are symmetrical to the y axis.
f(x) = 2 - (x-1)^2 is the upper curve, and g(x) = (x-1)^2 the lower one.
Use
V = 2pi∫(x(f(x) - g(f))dx

V = 2pi*∫(x*(2 - (x-1)^2 - (x-1)^2) dx from x = 0 to 2

V = 2pi*∫(x*(2 - 2(x-1)^2))dx

extend and integrate:

V = 16,7552

Regards

2pi∫(x(2-2(x-1)^2))dx
= 2pi[-1/2*x^4+4/3*x^3]