Use the method of cylindrical shells to find the volume generated by rotating the region bounded by y=4(x−2)2 and y=x2−4x+7 about the y-axis?

The common points on the x axis will be

4(x-2)^2= x^2 -4x +7= 4x^2 -16x +16

3x^2 -12x +9=0

x^2 - 4x+3=0 = (x-3)(x-1)

Limits are x=3 and x=1.

Now to find the volume, you need to integrate 2piy dx between x=3 and 1. If you select y=x^2-4x+3, You get the volume 2 pi 8/3. If you the other, you get volume = 2 pi 10/3. You need to select 2 pi 10/3 as the volume generated around y axis.