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`

`=pi [(x^7)/(7)+(x^4)/(2)+x]_0^3`

`=pi(|355.93|-|0|)`

`=1118.2\ text[units]^3`