Question

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

Answer 1

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