Calculus: Early Transcendentals, Chapter 6, 6.3, Section 6.3, Problem 7

The shell has the radius x , the cricumference is 2pi*x and the height is 6x - 2x^2 - x^2 , 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:
x^2 = 6x - 2x^2 =>3x^2 - 6x = 0 => 3x(x - 2) = 0 => 3x = 0 => x = 0 and x - 2 = 0 => x = 2
V = 2pi*int_0^2 x*(6x - 3x^2) dx
V = 2pi*(int_0^2 6x dx - int_0^2 3x^2 dx)
Using the formula int x^n dx = (x^(n+1))/(n+1) yields:
V = 2pi*(6x^2/2 - 3x^3/3)|_0^2
V = 2pi*(3x^2 - x^3)|_0^2
V = 2pi*(3*2^2 - 2^3)
V = 2pi*4
V = 8pi
Hence, evaluating the volume, using the method of cylindrical shells, yields V = 8pi.