## pappus

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

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.

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}
$

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