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.
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Those graphs cross at (2, 4) and (5, 19). Each cross section of the figure is a "washer" whose outer diameter is and the inner diameter is .
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.