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:

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