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Thread: Solid of revolution

  1. #1
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    Solid of revolution

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

    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 $\displaystyle 64\pi$ (if I should believe the textbook). Any ideas?


    EDIT: Sorry, corrected the typo.
    Last edited by atreyyu; Oct 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 $\displaystyle x = 3$ in quads I and IV between $\displaystyle y = 4-x^2$ , $\displaystyle y = 4$ , and $\displaystyle x = 3$ ... ???
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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 $\displaystyle y=4-x^2$ and $\displaystyle y=0$, around the axis $\displaystyle x=3$.
    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 $\displaystyle 64\pi$ (if I should believe the textbook). Any ideas?
    Make this substitution $\displaystyle x=x_1+3,~y=y_1$, then

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

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

    Now you can use the standard formula

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

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