Question

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

Question: Write two different iterated integrals for the double integral of \(f(x, y)\) over the rectangular region \(R\) defined by \(-2 \leq x \leq 4\) and \(1 \leq y \leq 5\). Answer: The two iterated integrals are: 1. \(\int_{1}^{5}\int_{-2}^{4}f(x,y)\,dx\,dy\) 2. \(\int_{-2}^{4}\int_{1}^{5}f(x,y)\,dy\,dx\)

Step-by-step solution

Iterated Integral: First Integrate with respect to x, then y

The first way we can set up the iterated integral is to integrate first with respect to \(x\) and then with respect to \(y\). In this case the limits of integration for \(x\) are \(-2 \leq x \leq 4\), and for \(y\) are \(1 \leq y \leq 5\). The iterated integral is given by: $$\int_{1}^{5}\int_{-2}^{4}f(x,y)\,dx\,dy$$

Iterated Integral: First Integrate with respect to y, then x

Alternatively, we can set up the iterated integral by integrating first with respect to \(y\) and then with respect to \(x\). In this case the limits of integration for \(y\) are \(1 \leq y \leq 5\), and for \(x\) are \(-2 \leq x \leq 4\). The iterated integral is given by: $$\int_{-2}^{4}\int_{1}^{5}f(x,y)\,dy\,dx$$ Both of these iterated integrals represent the same double integral \(\iint_{R} f(x, y) dA\) over the region \(R\).