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  1. #1
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    Solid of revolution

    Find the volume of solid of revolution generated by rotating the lamina contained between y=4-x^2 and y=0, around the axis x=3.

    I understand the premise of this task but can't quite arrive at the correct expression for integration, which should lead me eventually to the answer of 64\pi (if I should believe the textbook). Any ideas?


    EDIT: Sorry, corrected the typo.
    Last edited by atreyyu; October 10th 2011 at 03:36 PM.
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  2. #2
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    Re: Solid of revolution

    clarification, please ... is the region to be rotated about x = 3 in quads I and IV between y = 4-x^2 , y = 4 , and x = 3 ... ???
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  3. #3
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    Re: Solid of revolution

    Quote Originally Posted by atreyyu View Post
    Find the volume of solid of revolution generated by rotating the lamina contained between y=4-x^2 and y=0, around the axis x=3.
    I understand the premise of this task but can't quite arrive at the correct expression for integration, which should lead me eventually to the answer of 64\pi (if I should believe the textbook). Any ideas?
    Make this substitution x=x_1+3,~y=y_1, then

    y_1=4-(x_1+3)^2 and y_1=0, around the axis x_1=0.

    Find the roots y_1=4-(x_1+3)^2,~y_1=0, you must get x_1=-5 and x_1=-1

    Now you can use the standard formula

    V=-2\pi\int\limits_{-5}^{-1}x_1(4-(x_1+3)^2)\,dx_1=\ldots=64\pi~\mathsf{(cubic~units  )}

    Last edited by DeMath; October 13th 2011 at 05:12 AM.
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