The average value of f(x, y) = sin(x + y) over the rectangle R is -2/3π.
To find the average value of f(x, y) = sin(x + y) over the rectangle R = [0, π/6] × [0, 3π/2], we can use the formula for the average value of a function over a rectangle:
∫∫_R f(x, y) dA / |R|
where |R| is the area of the rectangle.
In this case, we have:
∫∫_R sin(x + y) dA / |R| = ∫₀^(π/6) ∫₀^(3π/2) sin(x + y) dx dy / [(π/6)(3π/2)]
To evaluate this double integral, we can use double integration by parts. First, we integrate with respect to x:
∫₀^(π/6) ∫₀^(3π/2) sin(x + y) dx dy = ∫₀^(π/6) [-cos(x + y)] ₀^(π/6) dy = ∫₀^(3π/2) cos(y) dy
Next, we integrate with respect to y:
∫₀^(π/6) cos(y) dy = sin(y) ₀^(3π/2) = -1
Therefore, the average value of f(x, y) = sin(x + y) over the rectangle R is:
∫∫_R sin(x + y) dA / |R| = -1 / [(π/6)(3π/2)] = -2/3π