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 = 12x − x^2, y = 32; about x = 4

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 = 12x − x², y = 32;  about x = 4

V=?

Answer 1

by shells

for dV=2π*r(x)*h(x)dx

r=x-4

h=-x²+12-32

the integrand is 2π*(-x³+16x²-80x+128)dx

the integral is from x=4 to x=8

volume is 2π*21