Final answer:
To find the average value of f(x,y)=sin(x + y) over the given rectangle, calculate the double integral of the function over the region and divide it by the area of the region.
Step-by-step explanation:
To find the average value of the function f(x,y)=sin(x + y) over the rectangle 0 < x < π/6, 0 < y < 3π/2, we need to calculate the double integral of f(x,y) over the given region and divide it by the area of the region.
1. First, find the integral of f(x,y) with respect to x: ∫sin(x + y) dx = -cos(x + y) + C
2. Next, substitute the limits of x into the equation from 0 to π/6:
o -cos(π/6 + y) + cos(0 + y)
o -cos(π/6 + y) + cos(y)
o -1/2cos(y + π/6) + cos(y)
3. Now, find the integral of the expression from the previous step with respect to y: ∫[-1/2cos(y + π/6) + cos(y)] dy = [-1/2sin(y + π/6) + sin(y)] + C'
4. Finally, substitute the limits of y into the equation from 0 to 3π/2:
o [-1/2sin(3π/2 + π/6) + sin(3π/2)] - [-1/2sin(0 + π/6) + sin(0)]
o [-1/2sin(5π/3) + sin(3π/2)] - [-1/2sin(π/6) + sin(0)]
Simplifying further will give you the average value of f(x,y) over the given rectangle.