Please instruct on finding the center of mass given the vector coordinates in double integral form stated. <=x<=sqrt(a)$ and $0<=y<=a-x^2$. $p(x,y)$ is a density function but it can be treated as a constant $c$. ... Please instruct on how to go about solving for the center of mass formula (doing the integrals). Thank you very much.

Start first by sketching out your feasible region. That is, the region bounded by, $y = a - x^2$ and $y = 0$ (noting $a$ is always positive).

You should get something that resembles a parabola with a maximum at $(0, a)$ and $x$ intercepts at $-\sqrt{a}$ and $\sqrt{a}$.

From there, you can choose your limits of integration appropriately. More specifically, choose (based on the graph) which values $x$ you want to integrate over and which values of $y$ you want to integrate over. Note that one integral should contain only constants.

For the denominator you should have,

$$\int_{-\sqrt{a}}^{\sqrt{a}}\int_{0}^{a - x^2} \ \rho(x,y) \ dy \ dx$$

Start by integrating the 'inside', $$\int_{0}^{a - x^2} \ \rho(x,y) \ dy\hspace{1cm}(1)$$

Then take what you computed in $(1)$ (we will call it $f$) and integrate it again, but this time wrt your limits of $x$,

$$\int_{-\sqrt{a}}^{\sqrt{a}} \ f \ dy\hspace{1cm}$$

Follow similar steps to compute the respective numerators.