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Thread: reduction,centroid,pappus

  1. #1
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    Joined
    Dec 2009
    Posts
    84

    reduction,centroid,pappus

    1) question asks to find reduction formual for

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

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


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


    3) given area of ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 $ is $\displaystyle \pi ab $ and that the volume generated when this area rotates through $\displaystyle \pi $ about the $\displaystyle x$-axis is $\displaystyle \frac{4}{3}\pi ab^2$
    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 $\displaystyle x$-axis,but the ellipse does
    "ignoring that"
    by pappus,with $\displaystyle G$ $\displaystyle y$-coord of centroid

    V=Axdistance travelled by G

    so
    $\displaystyle
    \frac{4}{3} \pi ab^2=(\pi ab)(\pi G)
    $

    $\displaystyle
    G=\frac{4b}{3\pi}
    $
    which is correct,but im worried about the crossing x-axis bit.
    Last edited by jiboom; Mar 12th 2010 at 09:54 AM.
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  2. #2
    Member
    Joined
    Dec 2009
    Posts
    84
    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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