`y = root(3)(x), y = 0, x = 1` Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given...

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.`