Question

Find the volume of the solid formed by rotating the region enclosed by
x=0 x=1 y=0 y=9+x3

Answer 1

What is the axis of rotation? I'll assume we're letting y = 0 be the axis of rotation. The volume of a solid is given by:

\[V = \int\limits_{a}^{b}A(x)dx\]

where A(x) is the area of a cross section made perpendicular to the x-axis. Since we are rotating, the cross section is a circle. Its radius is given by f(x) = 9 + x^3 - 0 = 9 + x^3.

\[A = \pi (9 + x^3)^2\]

We are finding the volume from x = 0 to x = 1, so

\[V = \pi \int\limits_{0}^{1}(9+x^3)^2dx\]

Using the fundamental theorem of calculus, you should find that the volume is (1199/14)*pi