Question

Solid of revolution problem.
I need to calculate the volume of the region bounded by y=x^3 and y=x, when x>= 0

Answer 1

Should I just substract the two curves and then calculate the volume with pi * integral of the result?

Answer 2

Simply subtracting the curves and integrating gives the area between the curves, scaled by a factor of \(\pi\). You're interested in the volume, which involves subtracting the squares of the curves.

Integrating via the washer method is essentially taking the sum of the areas of an infinite number of annuli (ring shapes).

The area of any one annulus is the difference between the area of the larger disk and the smaller disk.
\[A=\pi R^2-\pi r^2=\pi (R^2-r^2)\]