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

1 Answer
Jun 21, 2018

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.