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Math Help - Please help me solve the volume question about the x - axis

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    Please help me solve the volume question about the x - axis

    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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    Quote Originally Posted by shannon1111 View Post
    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.


    Thanks!~
    Those graphs cross at (2, 4) and (5, 19). Each cross section of the figure is a "washer" whose outer diameter is -x^2+ 12x- 16 and the inner diameter is 5x- 6.

    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 \pi r^2: \pi((-x^2+ 12x- 16)^2- (5x-6)^2). Taking the thickness of each cross section to be "dx", each cross section has volume \pi((-x^2+ 12x- 16)^2- (5x-6)^2)dx and "adding" (integrating) those cross sections gives the entire volume:
    \pi\int_2^5 ((-x^2+ 12x- 16)^2- (5x-6)^2)dx
    The simplest way to integrate that is to multiply it all out before integrating.
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