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Computer the Volume of a region bounded by 3 curves

  1. Sep 14, 2014 #1

    RJLiberator

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    1. The problem statement, all variables and given/known data
    Let R be the region in the first quadrant bounded by all three of the curves x = 2, y = 1, and y = (x−4)^2.
    Compute the volumes V1, V2, and V3 of the solids of revolution obtained by revolving R about the x-axis, the y-axis, and the x = 5 line, respectively.


    FIRST, I am trying to conceptualize this problem. I have the 3 necessary curves graphed. A 'triangle' looking figure is formed between y=1 to 4 and x =2 to 3. Do I need to find the volume of THIS figure revolved around the various axis points OR do I need to find the volume between y=0 to 1 and x = 2 to 4.

    Both areas are bounded by all 3 curves. My intuition tells me to take the volume of the 'triangle' looking figure, but I did not want to proceed until I figured this part out.
     
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  3. Sep 14, 2014 #2

    RJLiberator

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    **FIRST POST AND THIS POST ARE TWO SEPARATE PROBLEMS**

    The initial problem (previous problem) states:

    Let R be the region in the first quadrant bounded by y = 1 − x2. Compute
    the volume V of the solid of revolution generated by revolving R about the x-axis by using
    (a) slices
    and
    (b) shells.
    Please verify that you obtain the same value of the volume V by method (a) as by method (b).
    Use the calculation above to find the volume of the ball of radius 1 (in 3-dimensional space).

    Referring to the bold statements, wouldn't the volume of the radius 1 ball be equivalent to the volume of the region (what I had calculated for that problem) since the bounds are from 0 to 1?
     
  4. Sep 14, 2014 #3

    LCKurtz

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    Yes.

    Your first thought is correct. ##y=0## is not given as a boundary so your second interpretation is wrong.
     
  5. Sep 14, 2014 #4

    LCKurtz

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    You should start a new thread with a new problem.
     
  6. Sep 14, 2014 #5

    Zondrina

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    Here's a visual to help you conceptualize:

    Screen Shot 2014-09-14 at 4.36.04 PM.png

    The only region that is bounded by all three curves is the one depicted above.

    If you rotate this region about the x-axis, what is the volume element ##dV## you would choose to integrate?

    I believe choosing vertical slices will produce two integrals. Horizontal slices appear to be better as you will require only one integral.
     
  7. Sep 14, 2014 #6

    LCKurtz

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    That is not the correct area. Also you should quote at least part of the post to which you are replying.
     
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