Verified answer

Since x, y > 0, we can take the density as δ(x,y) = k|x| = kx for some constant k.

So, m = ∫∫ δ(x,y) dA

..........= ∫∫ kx dA

..........= ∫(θ = 0 to π/2) ∫(r = 0 to 6) k(r cos θ) * (r dr dθ), via polar coordinates

..........= k * ∫(θ = 0 to π/2) cos θ dθ * ∫(r = 0 to 6) r^2 dr

..........= k * (sin θ {for θ = 0 to π/2}) * ((1/3)r^3 {for r = 0 to 6})

..........= k * 1 * 72

..........= 72k.

My = ∫∫ x δ(x,y) dA

......= ∫∫ x * kx dA

......= ∫(θ = 0 to π/2) ∫(r = 0 to 6) k(r cos θ)^2 * (r dr dθ), via polar coordinates

......= k * ∫(θ = 0 to π/2) cos^2(θ) dθ * ∫(r = 0 to 6) r^3 dr

......= k * ∫(θ = 0 to π/2) (1/2)(1 + cos(2θ)) dθ * ∫(r = 0 to 6) r^3 dr

......= k * (1/2)(θ + sin(2θ)/2) {for θ = 0 to π/2}) * ((1/4)r^4 {for r = 0 to 6})

......= k * (π/4) * 6^4/4

......= 81πk.

Mx = ∫∫ y δ(x,y) dA

......= ∫∫ y * kx dA

......= ∫(θ = 0 to π/2) ∫(r = 0 to 6) (r sin θ) * k(r cos θ) * (r dr dθ), via polar coordinates

......= k * ∫(θ = 0 to π/2) sin θ cos θ dθ * ∫(r = 0 to 6) r^3 dr

......= k * (1/2) sin^2(θ) {for θ = 0 to π/2}) * ((1/4)r^4 {for r = 0 to 6})

......= k * (1/2) * 6^4/4

......= 162πk.

So the center of mass is (My/m, Mx/m) = (81πk/(72k), 162πk/(72k)) = (9π/8, 9π/4).

I hope this helps!