I have now sorted the first part of this question but cant get anywhere with part(b):

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.

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}

$

I am unable to do the next part of the question:

b) find the first moment about Ox of that part of the area for which y>=0

i seem to need the perimeter of the ellipse here?

the answer in the book is