`y = x^3, y = 0, x = 1, x = 2` 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 `x^3` , hence, the volume can be evaluated, using the method of cylindrical shells, such that:

`V = 2pi*int_1^2 x*(x^3) dx`

`V = 2pi*int_1^2 x^4dx`

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

`V = 2pi*x^5/5|_1^2`

`V = 2pi*(2^5 - 1^5)/5`

`V = (62*pi)/5`

Hence, evaluating the volume, using the method of cylindrical shells, yields `V = (62*pi)/5.`