Problem 37 - Calculus Early Transcendentals

Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The region of integration for \(\int_{4}^{6} \int_{1}^{3} 4 d x d y\) is a square. b. If \(f\) is continuous on \(\mathbb{R}^{2}\), then $$ \int_{4}^{6} \int_{1}^{3} f(x, y) d x d y=\int_{4}^{6} \int_{1}^{3} f(x, y) d y d x $$ c. If \(f\) is continuous on \(\mathbb{R}^{2}\), then $$ \int_{4}^{6} \int_{1}^{3} f(x, y) d x d y=\int_{1}^{3} \int_{4}^{6} f(x, y) d y d x $$

Step-by-Step Solution

Answer

Answer: The region of integration is a square. Changing the order of integration or the limits does not affect the result for a continuous function on the given rectangular region.

Step 1: Statement a)

Consider the given double integral: \(\int_{4}^{6} \int_{1}^{3} 4 d x d y\). The limits of integration for \(x\) are from 1 to 3, and for \(y\) are from 4 to 6. Now, let's find the region of integration. To do so, we need to look at the intervals where \(x\) and \(y\) are defined. \(x\) is in the interval [1,3] and \(y\) is in the interval [4,6]. This defines a rectangle with sides of length 2 (3-1) and 2 (6-4). Since the sides are of equal length, the region of integration is a square.

Step 2: Statement b)

We are given a continuous function \(f\) on \(\mathbb{R}^{2}\) and we want to determine if we can change the order of integration without affecting the result. In this case, the integral is already in the form \(\int_{4}^{6} \int_{1}^{3} f(x, y) d x d y\). We want to understand if it is equal to the integral with the order of the differentials switched: \(\int_{4}^{6} \int_{1}^{3} f(x, y) d y d x\). Recall that a continuous function on a closed interval is always integrable. Moreover, Fubini's theorem states that if a function \(f\) is continuous on a rectangular region, then the iterated integrals are equal, meaning: $$\int_{a}^{b} \int_{c}^{d} f(x,y) dxdy = \int_{a}^{b} \int_{c}^{d} f(x,y) dydx$$ So, in this case, the statement is true.

Step 3: Statement c)

We are given a continuous function \(f\) on \(\mathbb{R}^{2}\) and we want to determine if we can switch the limits of integration without affecting the result. In this case, the integral is already in the form \(\int_{4}^{6} \int_{1}^{3} f(x, y) d x d y\). We want to understand if it is equal to the integral with the order of the limits swapped: \(\int_{1}^{3} \int_{4}^{6} f(x, y) d y d x\). Again, Fubini's theorem states that if a function \(f\) is continuous on a rectangular region, then the iterated integrals are equal. Therefore, we can change the order of integration without affecting the result: $$\int_{4}^{6} \int_{1}^{3} f(x,y) dxdy = \int_{1}^{3} \int_{4}^{6} f(x,y) dydx$$ So, in this case as well, the statement is true.