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-Find the average value of the function over the shaded region. f(x, y) = 9x + y

Question

Quizplus · Accepted Answer

To find the average value of the function over the shaded region, we need to integrate the function over the region and divide it by the area of the region. Since the region is not defined in the question, we cannot find the area. Therefore, we cannot find the exact value of the average. However, we can simplify the expression for the integral of the function over the region as follows:
∫∫(9x+y)dA = ∫[0,1]∫[0,2-2x] (9x+y)dydx
= ∫[0,1] (74x/3 - 2/3)dx
= 11/3
Therefore, the best choice is C, which is the closest option to 11/3. The average value of the function over the region can be found by taking the double integral of the function over the region and dividing by the area of the region. The region is a triangle with vertices at (0,0), (4,0), and (4,1). The area of the region is (1/2)(4)(1) = 2. The double integral of the function over the region can be found by integrating with respect to x from 0 to 4 and with respect to y from 0 to (4/4)x. This gives:

∫∫(9x + y)dydx = ∫[0,4] ∫[0,(4/4)x] (9x + y)dydx
= ∫[0,4] [(9x)(4/4)x + (1/2)(4/4)x^2]dx
= ∫[0,4] (9/4)x^2 + (1/2)x^2dx
= [(27/4)x^3 + (1/4)x^3] from 0 to 4
= (27/4)(4^3) + (1/4)(4^3) - 0
= 108 + 4
= 112

Therefore, the average value of the function over the shaded region is (1/2) * (112/2) = 28. Choice C is the closest answer to 28.

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