Question

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis.
y = 6x
y = 18
x = 0

Answer 1

V = Int(2pi*r*h dx)[0,3]
where
r = x
h = 18-y

V = 2pi*Int(x*(18-6x) dx)[0,3]
= 2pi * Int(18x - 6x^2 dx)[0,3]
= 2pi*(9x^2 - 2x^3)[0,3]
= 2pi*(27-18)
= 18pi