1. ## reduction,centroid,pappus

1) question asks to find reduction formual for

$
I_(n)=\int \frac{sin(2nx)}{sin(x)}
$

then it asks hence or otherwise find:
$
\int_0^{\frac{\pi}{2}} \frac{sin(5x)}{sin(x)}
$

only way i can do this is to do a reduction formula for

$
\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 $x$-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
$
\frac{4}{3} \pi ab^2=(\pi ab)(\pi G)
$

$
G=\frac{4b}{3\pi}
$

which is correct,but im worried about the crossing x-axis bit.

2. 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)