Answer :
Answer: The center of mass =
(8192/(1280π), 8192/(1280π)).
Step-by-step explanation:
Here, δ(x,y) =k(x^2 + y^2) for some constant k.
So, m = ∫∫ δ(x,y) dA
..........= ∫(θ = 0 to π/2) ∫(r = 0 to 4) kr^2 × (r dr dθ), via polar coordinates
..........= (π/2) × (k/4)r^4 {for r = 0 to 4}
..........= 256πk/8.
My = ∫∫ x δ(x,y) dA
......= ∫(θ = 0 to π/2) ∫(r = 0 to 4) (r cos θ) × kr^2 × (r dr dθ), via polar coordinates
......= ∫(θ = 0 to π/2) cos θ dθ × ∫(r = 0 to 4) kr^4 dr
......= sin θ {for θ = 0 to π/2} × (1/5)kr^5 {for r = 0 to 4}
......= 1024k/5.
Mx = ∫∫ y δ(x,y) dA
......= ∫(θ = 0 to π/2) ∫(r = 0 to 4) (r sin θ) × kr^2 × (r dr dθ), via polar coordinates
......= ∫(θ = 0 to π/2) sin θ dθ × ∫(r = 0 to 4) kr^4 dr
......= -cos θ {for θ = 0 to π/2} × (1/5)kr^5 {for r = 0 to 4}
......= 1024k/5.
Hence, the center of mass is (My/m, Mx/m) =
My/m = 1024/5 ×8/256
The same for Mx/m the density at any point is proportional to the square of its distance from the origin
(8192/(1280π), 8192/(1280π)).