Use the method of cylindrical shells to find the volume v generated by rotating the region bounded by the given curves about the specified axis. y = 16 x, y = 0, x = 1; about x = ?2

First let's move change the coordinate system so that the rotation axis becomes y

then f (x) = '16x becomes f (x) = 16 (x-2) = 16x-32, y=0 stays as it is but x=1 becomes x=3

then we apply the formula for cyl shells method:

V=integral 2 to 3 (2pi*x*f (x)) dx = int 2-3 (2pi*x*16x-2pi*x*32) dx=

int 2-3 32pi*x^2 dx - int 2-3 64pi*x dx = 2-3| 32pi*x^3/3 - 32pi*x^2 =

32pi*27/3 - 32pi*9 - 32pi*8/3 + 32pi*4 = 32pi * (9-9-8/3+4) = 32pi*4/3 = 128pi/3