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Volume of rotation, y-axis

  1. May 1, 2014 #1
    1. The problem statement, all variables and given/known data

    Hello!

    English is not my native language so I hope the terminology is right.

    Q:
    Find the volume generated by the curve y=1/x+2, positive x- and y-axis and the line x=1.
    Calculate the volume obtained by rotation around the:
    a) x-axis
    b) y-axis

    2. Relevant equations
    The text book use this one:
    [tex]Vx= \pi \int_{a}^{b}(f(x))^{2} dx[/tex]

    3. The attempt at a solution
    a) I got this one right:

    [tex]Vx = \pi \int_{0}^{1}\frac{1}{(x+2)^{2}} dy= \pi \left[ -\frac{1}{x+2} \right]= \pi(\frac{1}{3} - \frac{1}{2}) = \frac{\pi}{6}[/tex]

    b) I can't get this straight and I'm not sure about the upper-/lower-limits:

    [tex] y(1)= \frac{1}{1+2}=\frac{1}{3}\\y(0)=\frac{1}{0+2}=\frac{1}{2}.[/tex]

    X as a function of y:

    [tex]x(y)=\frac{1}{y}-2[/tex]

    We have:

    [tex]x(y)=\frac{1}{y}-2\\Vy = \pi \int_{1/2}^{1/3}(\frac{1}{y}-2)^{2} dy = 4\pi \left[y-\frac{1}{4y}-lny \right] = 4\pi ((\frac{1}{2}-\frac{1}{2}-ln\frac{1}{2}) - (\frac{1}{3}-\frac{3}{4}-ln\frac{1}{3}))=\\= 4\pi (-\frac{1}{3}+\frac{3}{4}+ln\frac{1}{3}-ln\frac{1}{2}) = 4\pi (\frac{5}{12}+ln\frac{2}{3}) [/tex]

    Stock here! I have tested the limits 0 to 1, 0 to 1/3 och 1/3 to ½. Anybody have any clue?



    The answer is:

    [tex]4\pi (\frac{1}{2}+ ln\frac{2}{3}) v.e[/tex]
     
  2. jcsd
  3. May 1, 2014 #2
    I would have to say that the shell method would be easier here,
    2*∏*∫(SHELL RADIUS)*(SHELL HEIGHT) dx
    Of course the limits of integration would be [0,1]
     
  4. May 1, 2014 #3

    Zondrina

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    Homework Helper

    It would be convenient to use vertical line segments in this case in order to avoid setting up multiple integrals. So:

    ##r_{in} = x - dx##
    ##r_{out} = x##
    ##height = \frac{1}{x+2}##

    ##dV = 2\pi(\frac{r_{in} + r_{out}}{2})(r_{out} - r_{in})(height)##
     
  5. May 1, 2014 #4
    Thanks,
    I have never seen this methods. The text book & teacher only use pi(f(x))² and Washer method. Should it not be possiable with disc/washer?

    With [0,1] I got:
    [tex]Vy = 4\pi ((1-\frac{1}{4*1}-ln1) - (\frac{0}{0}-\frac{1}{4*0}-ln\frac{0}{0})) = 4\pi (1-\frac{1}{4}) = 4\pi (3/4) [/tex]
    Isn't the parentheses with 0 undefined, so I can't really use lower limit 0?
     
  6. May 1, 2014 #5

    Zondrina

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    Homework Helper

    The method I'm using is simply a generalized version of the disk and washer methods. Makes it so you only have to remember one formula really.

    Using the method I get 1.18796 as the answer.
     
  7. May 1, 2014 #6
    Using the shell method I got the same answer as Zondrina, but using your integral Hacca, I evaluated the integral on my own, and on wolfram, and both came to the same conclusion of 0.0448.
     
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