Question

.Find the average value of 𝑓(𝑥, 𝑦) = 𝑥

2𝑦 over the region 𝑅 which is a rectangle with vertices

(−1, 0), (−1, 5), (1, 5), (1, 0).

Answer 1

$R=[a,b]×[c,d]=[-1,1]×[0,5]\\ f_{avg}=\frac{1}{(b-a)(d-c)}\int_c^d\int_a^b f(x,y)dxdy\\ =\frac{1}{(1-(-1))(5-0)}\int_0^5\int_{-1}^1x^2ydxdy\\ =\frac{1}{10}\int_0^5 \frac{1}{3}[x^3] _{-1}^1ydy\\ =\frac{1}{30}\frac{1}{2}[y^2]_0^5\\ =\frac{1}{60}×25\\ =\frac{5}{12}$