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.