Find the volume when the area bounded by f(x) = xe^x, y = e, and the y-axis, is revolved around the y-axis.

Answer 1

xe^x = e when x = 1

using shells,
v = Int(2*pi*r*h dx)[0,.852]
where r = x
h = e-y = e - xe^x

v = 2pi*Int(x(e-xe^x) dx)
= 2pi *Int(ex - x^2 e^x) dx
= 2pi * (e/2 x^2 - e^x (x^2 - 2x + 2))[0,1]
= pi(4-e)
= 4.03