Question

Consider the region R in ℝ² such that for any continuous function f: ℝ²→ℝ, the value of ∬ᵣ f d A can be expressed as the sum of two iterated integrals

Given ∬_R f dA = ∫_(0 to 1) ∫_(0 to 2y) f(x,y) dx dy + ∫_(1 to 3) ∫_(0 to 3-y) f(x,y) dx dy and let f(x,y) be some continuous function on R². Set up a single iterated integral that computes ∬_R f dA.

Answer 1

To compute the given double integral as a single iterated integral, we can change the order of integration. By rearranging the limits of integration, we can set up a single iterated integral that represents the region R and computes the desired integral.

Step-by-step explanation:

To set up a single iterated integral that computes ∬R f dA, we can combine the two given iterated integrals by changing the order of integration. The given iterated integrals are:

∫01 ∫02y f(x,y) dx dy + ∫13 ∫03-y f(x,y) dx dy

By changing the order of integration, we can write a single iterated integral as follows:

∫03 ∫02y f(x,y) dy dx

This single iterated integral represents the area R and computes ∬R f dA.