This is the region as described, under a cubic curve.

123-151015202530xyOpen image in a new page

Area bounded by `y = x^3+1`, `x=0` and `x=3`.

When the shaded area is rotated 360° about the x-axis, we observe that a volume is generated:

123-151015202530-5-10-15-20-25-30xy3Open image in a new page

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`


`=1118.2\ text[units]^3`

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