`y = 4x - x^2, y = x` Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves...

The shell has the radius `x` , the cricumference is `2pi*x` and the height is `4x - x^2 - x` , hence, the volume can be evaluated, using the method of cylindrical shells, such that:

`V = 2pi*int_(x_1)^(x_2) x*(3x - x^2) dy`

You need to evaluate the endpoints `x_1` and `x_2` , such that:

`4x - x^2= x => 3x -x^2 = 0 => x(3 - x) = 0 => x = 0 and 3-x = 0 => x = 3`

`V = 2pi*int_0^3 x*(3x - x^2) dy`

`V = 2pi*(int_0^3 3x^2 dx - int_0^3 x^3dx)`

Using the formula `int x^n dx = (x^(n+1))/(n+1) ` yields:

`V = 2pi*(3x^3/3 - x^4/4)|_0^3`

`V = 2pi*(x^3 - x^4/4)|_0^3`

`V = 2pi*(3^3 - 3^4/4)`

`V = 2pi*(3^3)/4`

`V = (27pi)/2`

Hence, evaluating the volume, using the method of cylindrical shells, yields `V = (27pi)/2.`