Calculus

First of all, you want to be a bit more clear. The shape is revolved about the line x = 4. The boundaries of the shape being revolved are y = x^{3}, y = 0, and x = 2.

If you draw a diagram, you should see that beween using disks or cylindrical shells, the shells are the best choice.

the height of the region being revolved at position x is x^{3}. And that position is a distance 4 - x from the axis. Thus the volume of the shell is

dV = 2πrh dx = 2π(4 - x)x^{3} dx = 2π(4x^{3} - x^{4}).

The interval of integration is [0,2].

Thus the volume will be

2π∫_{0}^{2} (4x^{3} - x^{4}) dx

= 2π(x^{4} - x^{5}/5)|_{0}^{2} = 2π(16 - 32/5) = 96π/5