Question

Let d = ϕ(r), where ϕ(u, v) = (u^2, u v), and r = [3, 9] × [0, 4]. Calculate:

a) ∫∫ dy da

b) 108

c) 54

d) 27

Answer 1

To calculate ∫∫ dy da, we need to find the limits of integration for y and a. The limits for y are 0 to 4 and the limits for a are 3 to 9. Integrating da first, we get 4, and then integrating dy, we get 6. Therefore, the value of the integral is 24.

Step-by-step explanation:

To calculate ∫∫ dy da, we need to find the limits of integration for y and a. Given that r = [3, 9] × [0, 4], the limits for y are 0 to 4, and the limits for a are 3 to 9. So, the integral becomes: ∫∫ dy da = ∫∫[0,4] [3,9] dy da.

The area element dy represents the differential length in the y-direction, and the area element da represents the differential length in the x-direction. To integrate, we can reverse the order of integration, so the integral becomes: ∫[0,4] ∫[3,9] dy da = ∫[3,9] ∫[0,4] da dy.

Integrating da first, we get ∫[0,4] da = a|[0,4] = 4-0 = 4. Then, integrating dy, we get ∫[3,9] dy = y|[3,9] = 9-3 = 6. Therefore, the value of the integral is 6 * 4 = 24.