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 = x3, 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 x3. And that position is a distance 4 - x from the axis. Thus the volume of the shell is
dV = 2πrh dx = 2π(4 - x)x3 dx = 2π(4x3 - x4).
The interval of integration is [0,2].
Thus the volume will be
2π∫02 (4x3 - x4) dx
= 2π(x4 - x5/5)|02 = 2π(16 - 32/5) = 96π/5