Question

What is the surface area produced by rotating `f(x)=x^3-8, x in [0,2]` around the x-axis?

Answer 1

first find dS, note the radius of the rotation is x.
`dS=sqrt(1+9x^4)dx`
`int_0^2 2pixdS`
to solve make a substitution of `w=3x^2`

Explanation:

If you make the substitution with w your integral becomes
`int_0^12 pi/3*sqrt(1+w^2)dw`
with a trigonometric substitution `w=tan(theta)`
the integral becomes
`pi/3 int_0^arctan(12) sec^3(theta)d(theta)`
which you can solve by parts or using a table.