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Find the volume of the solid revolved around y=x^3, y=0, x=2 about the line x=4

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1 Answer

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