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Math Help - reduction,centroid,pappus

  1. #1
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    Dec 2009
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    reduction,centroid,pappus

    1) question asks to find reduction formual for

     <br />
I_(n)=\int \frac{sin(2nx)}{sin(x)}<br />
    then it asks hence or otherwise find:
     <br />
\int_0^{\frac{\pi}{2}} \frac{sin(5x)}{sin(x)}<br />
    only way i can do this is to do a reduction formula for

     <br />
\int_0^{\frac{\pi}{2}} \frac{sin(nx)}{sin(x)}<br />
    What am i missing to be able to use the first result?


    2) a surface of revolution is formed by rotating completely about the x-axis the arc of
    x=at^2,y=2at
    from t=0 to  t=\sqrt{3}
    denote by S the surface area,show that x', the x co-ord of the centroid of this surface is given by
    Sx'=8\pi a^3\int_0^3 t^3\sqrt{1+t^2} dt.


    3) given area of ellipse  \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 is \pi ab and that the volume generated when this area rotates through \pi about the x-axis is \frac{4}{3}\pi ab^2
    use pappus' theorem to find the centroid.
    for this i get the answer but not sure if its ok.


    pappus relates to area A enclosed by curve that does not cross x-axis,but the ellipse does
    "ignoring that"
    by pappus,with G y-coord of centroid

    V=Axdistance travelled by G

    so
     <br />
\frac{4}{3} \pi ab^2=(\pi ab)(\pi G)<br />

     <br />
G=\frac{4b}{3\pi} <br />
    which is correct,but im worried about the crossing x-axis bit.
    Last edited by jiboom; March 12th 2010 at 09:54 AM.
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  2. #2
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    Dec 2009
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    i have now sorted out questions 1 and 3.

    2 still eludes me. I can not get the t^3 in the integral and cant see why we have a 3 as the upper limit for the integral and not rt(3)
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