Find the average value of the function f(x,y) over the plane region R. f(x, y) = 6x2y3; R = {(x, y) | 0 x 2;

Find the average value of the function f(x,y) over the plane region R.
f(x, y) = 6x^2y^3; R = {(x, y) | 0 les than or equal to x less than or equal to 2; 1 less than or equal to y less than or equal to 4}

First of all, the area of R equals (2 - 0)*(4 - 1) = 6.

So, the average value of f = 6x^2 y^3 on R equals
(1/6) * ∫(x = 0 to 2) ∫(y = 1 to 4) 6x^2 y^3 dy dx; note that we divide the integral by the area of R
= ∫(x = 0 to 2) ∫(y = 1 to 4) x^2 y^3 dy dx
= ∫(x = 0 to 2) (1/4) x^2 y^4 {for y = 1 to 4} dx
= ∫(x = 0 to 2) (255/4) x^2 dx
= (255/4) * (1/3)x^3 {for x = 0 to 2}
= 170.

I hope this helps!

F=<_y