The region R enclosed by the curves y=5x−6 and y=−x2+12x−16 is rotated about the x-axis. Find the volume of the resulting solid.
The area of each such "washer" is the area of the outer circle minus the area of the inner circle, and each of those is : . Taking the thickness of each cross section to be "dx", each cross section has volume and "adding" (integrating) those cross sections gives the entire volume:
The simplest way to integrate that is to multiply it all out before integrating.