This is the region as described, under a cubic curve.
When the shaded area is rotated 360° about the x-axis, we observe that a volume is generated:
Area under the curve `y=x^3+1` from `x=0` to `x=3` rotated around the `x`-axis, showing a typical disk.
Applying the formula for the solid of revolution, we get
`V=pi int_a^b y^2 dx`
`=pi int_0^3(x^3+1)^2 dx`
`=pi int_0^3(x^6+2x^3+1) dx`