Question

4 0K,Os3l slaw Ht

show the steps please its emergency

Answer 1

f(x, y) = 1/4 + x/2 + y/2 + xy 0 lessthanorequalto x lessthanorequalto 1, 0 lessthanorequalto y lessthanorequalto 1 (a) integral^1_x = 0 integral^1_y = 0 1/4 + x/2 + y/2 xy dx dy = integral^1_x = 0 (1/4 + x/2) y Vertical-bar ^1_y = 0 + (x + 1/2) y^2/2 Vertical-bar ^1_y = 0 dx = integral^1_x = 0 1/4 + x/2 + x/2 + 1/4 dx = integral^1_x = 0 1/2 + x dx = x/2 + x^2/2 Vertical-bar ^1_x = 0 = 1 Hence proved. (b) f_x(x) = integral^infinity_-infinity f(x, y) dy = integral^1_y = 0 (1/4 + x/2 + y/2 + xy) dy = (1/4 + x/2) middot y Vertical-bar ^1_y = 0 + (1/2 + x) y^2/2 Vertical-bar ^1_y = 0 = 1/4 + x/2 + x/2 + 1/4 = 1/2 + x f_y(y) = integral^infinity_-infinity f(x, y) dx = integral^1_x = 0 (1/4 + x/2 + y/2 + xy) dx = (x^2/4 + x/4) Vertical-bar ^1_x = 0 + x middot y/2 Vertical-bar ^1_x = 0 + y x^2/2 Vertical-bar ^1_x = 0 f_y(y) = 1/2