Cylindrical Shells?

Question

I've tried so many times and the problem is so simple...

Consider the given curves to do the following:

y = 2 - (x^2)
y = x^2

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about x = 1.

Answer 1

It really helps to draw a diagram. It should show the region bounded by the parabolas, indicate the axis of revolution, and a representative "slice" of the region that, when rotated around the axis, generates a cylindrical shell. In this case it's a vertical "slice", located at some x, of width dx, whose top touches the parabola y = 2 - x² and whose bottom touches the parabola y = x².

The element of volume dV is given by

dV = 2πrh dx where

r is the distance from the axis of revolution to the "slice" -- here, it is 1 - x
h is the length of the "slice" -- here, it is (2-x²) - x², or 2 - 2x².

So dV = 2π(1 - x)(2 - 2x²) dx

and

V = ∫ 2π(1 - x)(2 - 2x²) dx

To find the limits for x, you need to determine the intersection points of the parabolas that define the region. The diagram is helpful for seeing this too.

You should get V = 16π/3