# reduction,centroid,pappus

• Mar 8th 2010, 01:26 PM
jiboom
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.
• Mar 10th 2010, 06:42 AM
jiboom
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)