Calculus: Early Transcendentals, Chapter 6, 6.3, Section 6.3, Problem 3

The shell has the radius x , the cricumference is 2pi*x and the height is root(3)(x) , hence, the volume can be evaluated, using the method of cylindrical shells, such that:
V = 2pi*int_(x_1)^(x_2) x*(6x - 3x^2) dx
You need to evaluate the endpoints x_1 and x_2 , such that:
root(3) x = 0 => x = 0^3 =>x = 0
V = 2pi*int_0^1 x*(root(3) x) dx
V = 2pi*(int_0^1 x^(1/3) dx
Using the formula int x^n dx = (x^(n+1))/(n+1) yields:
V = 2pi*(3/4)*(x^(4/3))|_0^1
V = (3pi/2)*(1^(4/3) - 0^(4/3))
V = (3pi)/2
Hence, evaluating the volume, using the method of cylindrical shells, yields V = (3pi)/2.