Use the shell method to find the volume of the solid generated by revolving the plane region about the line x = 9.

Question

y = 7x − x2
y = 0

Answer 1

V = Int(2pi*r*h dx)[0,7]
where
r = 9-x
h = y
V = 2pi*Int((9-x)(7x-x^2)dx)[0,7]
= 2pi*Int(x^3 - 16x^2 + 63x)dx [0,7]
= 2pi(1/4 x^4 - 16/3 x^3 + 63/2 x^2)[0,7]
= 2pi(2401/4 - 5488/3 + 3087/2)
= 2pi * 3773/12
= 3773/6 pi