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Math Help - Volume by rotation

  1. #1
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    Volume by rotation

    Please help! I am beating my head over this one.

    "
    A triangle is formed by the lines x = 1; x + y = 7; and
    3x-4y=7. Rotate the triangle about the line x=-1 to create a volume. Find volume by setting up definite integral.

    Thanks out there to anyone who can help.


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  2. #2
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    Hey, how about the washer method? Can you use the figure below and the following to do it? Pretty sure that's it. Did it kinda' quick. Double-check it ok.

    V=\pi\int_{-14}^{1}\left[ (f+1)^2-(g+1)^2\right] dy+\pi\int_1^6 \left[(f+1)^2-4\right] dy

    f and g are the equations for the two lines in terms of y right? For example f(y)=(7-y).
    Attached Thumbnails Attached Thumbnails Volume by rotation-washermethod.jpg  
    Last edited by shawsend; September 15th 2008 at 05:08 AM.
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  3. #3
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    Thanks, that looks right.
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  4. #4
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    after looking at it further, I figured that the +1 in the figure is the extra length on the radius b/c of rotation around the line x=-1. Did I figure right? Also, where did the negative four come from? Does it come from the inner cylinder formed with a volume of 4? Thanks again.
    Last edited by michaelaparker; September 15th 2008 at 07:15 AM.
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  5. #5
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    Quote Originally Posted by michaelaparker View Post
    after looking at it further, I figured that the +1 in the figure is the extra length on the radius b/c of rotation around the line x=-1. Did I figure right? Also, where did the negative four come from? Does it come from the inner cylinder formed with a volume of 4? Thanks again.
    The extra 1 comes from the rotation about a line 1 unit from the y-axis. Also, the 4 comes from the radius squared of the red line in the plot below revolved around the line x=-1.
    Attached Thumbnails Attached Thumbnails Volume by rotation-vor2.jpg  
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