Calculus: Early Transcendentals, Chapter 6, 6.3, Section 6.3, Problem 5

The shell has the radius x, the cricumference is 2pi*x and the height is e^(-x^2) , hence, the volume can be evaluated, using the method of cylindrical shells, such that:
V = 2pi*int_0^1 x*e^(-x^2) dx
You need to use substitution method to solve the integral, such that:
-x^2 = u => -2xdx = du => xdx = -(du)/2
V = 2pi*int_(u_1)^(u^2) e^u*(-du)/2
V = -pi*e^u|_(u_1)^(u^2)
V = -pi*e^(-x^2)|_0^1
V = -pi*(e^(-1^2) - e^(-0^2))
V = -pi*(1/e - e) => V = pi*(e - 1/e)
V = ((e^2-1)*pi)/e
Hence, evaluating the volume, using the method of cylindrical shells, yields V = ((e^2-1)*pi)/e.