Find the first moment, about the y-axis, of the arc of the parabola $\displaystyle y^2=1-x$ for which $\displaystyle x\geq o$. Find also the length if the arc and hence find the centroid of the arc. Use Pappus' first theorem to find the area of the surface generated when the arc rotates completely around the y-axis.

Moments is one of the things i don't exactly understand that well. I am given the formula $\displaystyle M=\sum y \delta x$ for the moment about the x-axis.

So I concluded that $\displaystyle M=\sum x\delta y$ for the moment about the y-axis.

$\displaystyle M=\displaystyle \int ^1_{-1} (1-y^2) \sqrt{1+4y^2} dy$

Solving this, I get: $\displaystyle \frac{7\sqrt{5}}{16}+\frac{17}{32}arsinh 2$. Answers given for this question are: 2; $\displaystyle \pi$' $\displaystyle \frac{2}{\pi}$; $\displaystyle 4\pi$.

Thanks!