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Thread: Volumes by Cylindrical Shells

  1. #1
    Senior Member mollymcf2009's Avatar
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    Volumes by Cylindrical Shells

    Hi everyone!

    Could someone show me a graph of this so I can verify mine? Also, I can't figure out if I need u-substitution on this or not. I've tried to work it both ways and am not getting the correct answer.

    Consider the given curves:

    $\displaystyle x = 8 + (y-3)^2$ and $\displaystyle x=9$

    Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis.


    Thanks!!!
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    Attachment 9801

    Your integral should start off looking like $\displaystyle 2\pi\int_2^4(9-(8+(y-3)^2))ydy$ and you should end up with $\displaystyle 8\pi$.
    I would suggest just multiplying out everything and then integrating term by term.
    Attached Thumbnails Attached Thumbnails Volumes by Cylindrical Shells-graphlik.jpg  
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  3. #3
    Senior Member mollymcf2009's Avatar
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    Beautiful! Thank you so much!!
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    Hello, mollymcf2009!

    Given the region bounded by: .$\displaystyle x \:= \:8 + (y-3)^2\:\text{ and }\:x\,=\,9$

    Use the method of cylindrical shells to find the volume of the solid
    obtained by rotating the region about the $\displaystyle x$-axis.

    We have a parabola: .$\displaystyle x \:=\:(y-3)^2 + 8$
    . . The vertex is $\displaystyle (8,3)$ and it opens to the right.
    $\displaystyle x = 9$ is a vertical line.

    The graph looks like this:
    Code:
          |                   |     *
          |                   *
          |               *:::|
          |            *::::::|
          |          *::::::::|
          |    (8,3)*:::::::::|
          |          *::::::::|
          |            *::::::|
          |               *:::|
          |                   *
          |                   |     *
      - - + - - - - - - - - - + - - - - - - -
          |                   9

    The curves intersect at $\displaystyle (9,2) \text{ and }(9,4)$


    The formula is: .$\displaystyle V \;=\;2\pi \int^b_a y\bigg[x_{\text{right}} - x_{\text{left}}\bigg]\,dy $

    So we have: .$\displaystyle V \;=\;2\pi\int^4_2y\bigg(9 - \left[8-(y-3)^2\right]\bigg)\,dy $

    . . and we must evaluate: .$\displaystyle V \;=\;2\pi \int^4_2\left(y^2 - 6y + 10\right)\,dy $

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