Final answer:
The average value of the function f(x, y) = x + a over the specified rectangle is 9 + 3a.
Step-by-step explanation:
The student is asking to find the average value of the function f(x, y) = x + a over a rectangle defined by the ranges 0 ≤ x ≤ 3 and 0 ≤ y ≤ 6. The average value of a function over a rectangle in 2D can be found by integrating the function over the area and then dividing by the area of the rectangle. In this case, the area of the rectangle is 3 × 6 = 18 square units.
The integral of f(x, y) = x + a with respect to x from 0 to 3 is ∫ (3x + 3a)dx which simplifies to 9x + 9a evaluated from 0 to 3, giving us 27 + 9a. We now integrate this with respect to y from 0 to 6 which is just multiplied by 6 since there are no y terms in our function, resulting in 6(27 + 9a).
Dividing the result by the area of the rectangle, the average value of the function is ± ± ± (6(27 + 9a)) ÷ 18 which simplifies to 9 + 3a. Therefore, the average value of f(x, y) over the specified rectangle is 9 + 3a.