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Volume of revolution

  1. May 1, 2014 #1
    DSC_0138~3[1].jpg

    DSC_0139~2[1].jpg

    View attachment 69284 1. The problem statement, all variables and given/known data

    i have done the part a, for b , i use the key in the (circled part equation ) in to calculator .. my ans is also different form the ans given. is my concept correct by the way?

    2. Relevant equations



    3. The attempt at a solution
     

    Attached Files:

    Last edited: May 1, 2014
  2. jcsd
  3. May 1, 2014 #2

    HallsofIvy

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    One obvious point is that you are missing a factor of "[itex]\pi[/itex]". The area of a circle is [itex]\pi r^2= \pi y^2[/itex].
     
  4. May 1, 2014 #3
    after adding pi, my ans is 2.80.... the ans is 5.047746784, which part is wrong?
     
  5. May 1, 2014 #4

    Zondrina

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    It would be a good idea here to use vertical line segments, otherwise you're going to have to set up multiple integrals. So leave everything as ##y(x)##, then:

    ##r_{in} = 0##
    ##r_{out} = 1 + \frac{1}{4x^2 + 1}##
    ##height = dx##

    ##dV = 2\pi(\frac{r_{in} + r_{out}}{2})(r_{out} - r_{in})(height)##

    Integrating the volume element should give you the answer you want.
     
  6. May 1, 2014 #5
    delsoo, your method is fine and I seem to get the same definite integral as you (which gives the correct answer too). The definite integral you have to evaluate is:

    $$\pi\int_0^{1/2} \left(1+\frac{1}{4x^2+1}\right)^2\,dx$$

    If you drop the factor of ##\pi##, you should get 1.60675.

    Use the substitution ##2x=\tan\theta## to make things easier.
     
  7. May 1, 2014 #6

    Zondrina

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    Erm this is misleading.

    The answer is indeed 5.04775 complements of wolfram:

    http://www.wolframalpha.com/input/?i=integrate+2pi%28+%281%2B+1%2F%284x^2%2B1%29%29%2F2+%29%281%2B+1%2F%284x^2%2B1%29%29+from+0+to+1%2F2
     
  8. May 1, 2014 #7
    Can you please explain to me how my statements are misleading? :)
     
  9. May 1, 2014 #8

    Zondrina

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    Your integrand is fine, it's just the answer you got I was worried about.
     
  10. May 1, 2014 #9
    I must be missing something but what is the problem with the answer I wrote? Are you talking about "1.60675"? :confused:
     
  11. May 1, 2014 #10

    Zondrina

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    Yeah I wasn't sure why you wrote that.
     
  12. May 1, 2014 #11
    Ah, I think I worded it poorly. What I meant was this:
    $$\int_0^1 \left(1+\frac{1}{4x^2+1}\right)^2\,dx=1.60675$$
    And I feel delsoo did some mistake while evaluating the above definite integral.
     
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