You're trying to find the mass of the region, so a triple integral is probably more appropriate here.
[tex]\displaystyle\text{mass}_S=\iiint_S\rho(x,y,z)\,\mathrm dV=\int_0^4\int_0^{\sqrt{16-x^2}}\int_0^{\sqrt25-x^2} xy\,\mathrm dz\,\mathrm dy\,\mathrm dx=\frac{706}5[/tex]
