A thin plate with constant mass per unit area$\displaystyle \rho$has edges defind by the curves,

$\displaystyle x = \sqrt{a^{2}-y^{2}}, y=x, y=0,$

where a>0 is a constant. Find the first moment of the plate about the x-axis.

See figure attached for my attempt. I've draw a quick sketch in the top right of the page.

I'm first calculating the total mass of the plate, then I can find the first moment of the entire plate about the x-axis.

i.e. $\displaystyle M\bar{y} = \int \int_{R}y \rho dA$

I'm going to first solve M.

Is there integral I've set up correct?

The one step I'm worried about is when I was left to integrate,

$\displaystyle \sqrt{a^{2}-y^{2}}$

if I had done that one correctly.

Is there perhaps another route of integrations that I should proceed with? Can anyone spot any other errors?

Thanks again!