A plate occupies the rectangle (0,4)X(0,3), and the density of the plate at (x,y) is equal to 8+x-y?

1) m = ∫(x = 0 to 4) ∫(y = 0 to 3) (8 + x - y) dy dx

........= ∫(x = 0 to 4) (8y + xy - y^2/2) {for y = 0 to 3} dx

........= ∫(x = 0 to 4) (39/2 + 3x) dx

........= (39x/2 + (3/2)x^2) {for x = 0 to 4}

........= 102.

2) Mx = ∫(x = 0 to 4) ∫(y = 0 to 3) x * (8 + x - y) dy dx

..........= ∫(x = 0 to 4) ∫(y = 0 to 3) (8x + x^2 - xy) dy dx

..........= 220, as above

My = ∫(x = 0 to 4) ∫(y = 0 to 3) y * (8 + x - y) dy dx

......= ∫(x = 0 to 4) ∫(y = 0 to 3) (8y + xy - y^2) dy dx

......= 144.

3) The center of mass is at (Mx/m, My/m) = (220/102, 144/102) = (110/51, 24/17).

I hope this helps!