Final answer:
To find the average value of the function, we need to calculate the double integral of the function over the triangular region and then divide it by the area of the region. The total value over the region can be found by solving the double integral, and then dividing it by the area to get the average value.
Step-by-step explanation:
To find the average value of the function, we need to calculate the double integral of the function over the triangular region and then divide it by the area of the region. The region is defined by the vertices (0, 0), (0, 2), and (2, 2).
Let's set up the double integral:
A = ∫∫_R f(x,y) dA
= ∫_0^2 ∫_0^{2-y} (-x+1) dx dy
Solving this integral will give us the total value over the region:
A = ∫_0^2 (1-y) (2-y) dy
By evaluating this integral, we can find the area of the region. Once we have the area, we divide the total value by the area to get the average value of the function over the region.