Final answer:
To evaluate the double integral ∬R(16−8y)dA, we first need to identify it as the volume of a solid. The region R is defined as R=[0,1]×[0,1], which represents a square with side length 1. By integrating the height (16−8y) over the region R, we can find the volume of the solid, which is equal to 12.
Step-by-step explanation:
To evaluate the double integral ∬R(16−8y)dA, we first need to identify it as the volume of a solid. The region R is defined as R=[0,1]×[0,1], which represents a square with side length 1. We can view the integrand (16−8y) as the height of the solid at each point (x, y) in R. By integrating the height over the region R, we can find the volume of the solid.
To evaluate the integral, we use the limits of integration given by R. So the double integral becomes:
∬R(16−8y)dA = ∫01 ∫01 (16−8y) dxdy.
Simplifying the integral, we have:
∫01 (16−8y) dy = [16y−4y2]
Substituting the limits of integration, we get:
[16(1)−4(1)2]−[16(0)−4(0)2] = 16−4 = 12.