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Chapter 16: Problem 3

Write two iterated integrals that equal \(\iint_{R} f(x, y) d A,\) where \(R=\\{(x, y):-2 \leq x \leq 4,1 \leq y \leq 5\\}\)

Short Answer

Expert verified
Answer: The two different iterated integrals are: 1) $$\int_{-2}^{4}\int_{1}^{5} f(x, y) \: dy \: dx$$ 2) $$\int_{1}^{5}\int_{-2}^{4} f(x, y) \: dx \: dy$$

Step by step solution

01

1. Find the first iterated integral (dy dx)

Given the region R: \(\{(x, y):-2 \leq x \leq 4,1 \leq y \leq 5\}\), we can determine the bounds for both dy and dx in the first iterated integral. The outer integral will be over x, and the inner integral will be over y. For x, the bounds are given: \(-2 \leq x \leq 4\). For y, it is also given: \(1 \leq y \leq 5\). The first iterated integral is: $$\int_{-2}^{4}\int_{1}^{5} f(x, y) \: dy \: dx$$
02

2. Find the second iterated integral (dx dy)

This time, we will switch the order of integration and first integrate over x and then over y. The outer integral will be over y, and the inner integral will be over x. The bounds for y are given: \(1 \leq y \leq 5\). The bounds for x are also given: \(-2 \leq x \leq 4\). The second iterated integral is: $$\int_{1}^{5}\int_{-2}^{4} f(x, y) \: dx \: dy$$ Now we have two iterated integrals representing the same double integral over the region R: 1) $$\int_{-2}^{4}\int_{1}^{5} f(x, y) \: dy \: dx$$ 2) $$\int_{1}^{5}\int_{-2}^{4} f(x, y) \: dx \: dy$$

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Most popular questions from this chapter

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