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 $$ \int_{4}^{6} \int_{1}^{3} f(x, y) d x d y=\int_{1}^{3} \int_{4}^{6} f(x, y) d y d x $$

Short answer

Question: Determine whether the following statements are true or false: a. The region of integration for the given double integral \(\int_{4}^{6} \int_{1}^{3} 4 d x d y\) is a square. b. For a continuous function \(f\) on \(\mathbb{R}^{2}\), \(\int_{4}^{6} \int_{1}^{3} f(x, y) d x d y=\int_{4}^{6} \int_{1}^{3} f(x, y) d y d x\). c. For a continuous function \(f\) on \(\mathbb{R}^{2}\), \(\int_{4}^{6} \int_{1}^{3} f(x, y) d x d y=\int_{1}^{3} \int_{4}^{6} f(x, y) d y d x\).

Step-by-step solution

Part A: Analyse the region of integration

Let's first understand the region of integration for the given integral \(\int_{4}^{6} \int_{1}^{3} 4 d x d y\). The region is described by \(4 \leq y \leq 6\) and \(1 \leq x \leq 3\). We can sketch a diagram to visualize the area and check whether it forms a square or not. The region of integration is a rectangle with vertices at (1,4), (3,4), (1,6), and (3,6), with lengths 3-1=2 along the x-axis and 6-4=2 along the y-axis. Since both lengths are the same, this region is indeed a square. Thus, the statement is true.

Part B: Evaluate the double integrals with different orders of integration

Given a continuous function \(f\) on \(\mathbb{R}^{2}\), we need to examine if the following equality holds: $$ \int_{4}^{6} \int_{1}^{3} f(x, y) d x d y=\int_{4}^{6} \int_{1}^{3} f(x, y) d y d x $$ Since \(f\) is a continuous function on \(\mathbb{R}^{2}\), we can apply Fubini's Theorem, which states that for continuous functions, the order of integration can be switched without changing the result. Therefore, the statement is true.

Part C: Evaluate the double integrals with respect to the same order of integration but different limits

Given a continuous function \(f\) on \(\mathbb{R}^{2}\), we need to examine if the following equality holds: $$ \int_{4}^{6} \int_{1}^{3} f(x, y) d x d y=\int_{1}^{3} \int_{4}^{6} f(x, y) d y d x $$ Notice that the order of integration is the same in both integrals (\(d x d y\)), but the limits of integration are switched. However, this does not change the result because the integration takes place over the same region. We can think of this as following the Commutative Property of addition \(a+b=b+a\) when we sum the function values over the region. Therefore, the statement is also true.

Key concepts

Region of Integration

When dealing with double integrals, the region of integration is crucial for determining the integral's limits. The region of integration can be visualized as the shape in a coordinate plane that you're integrating over. In the example exercise, the integral \( \int_{4}^{6} \int_{1}^{3} 4 \, dx \, dy \) is examined. This integral is spread over a rectangular region defined by the limits: \( 4 \leq y \leq 6 \) and \( 1 \leq x \leq 3 \).

You can sketch this region on the xy-plane, creating a rectangle with vertices at points \((1,4), (3,4), (1,6), \text{and} (3,6)\). Both the width and height of this rectangle measure 2 units. In this case, since the length across both axes is equal, the rectangular region of integration is actually a square.

Identifying the correct region of integration is necessary for ensuring correct integration and influences the final result of your problem.

Fubini's Theorem

Fubini's Theorem is a fundamental result in calculus that allows us to interchange the order of integration for double integrals when integrating continuous functions. This is particularly useful when solving problems involving double integrals, as sometimes changing the order can simplify calculations.

In the example, the statement \( \int_{4}^{6} \int_{1}^{3} f(x, y) \, dx \, dy = \int_{4}^{6} \int_{1}^{3} f(x, y) \, dy \, dx \) is evaluated. By Fubini's Theorem, if \( f \) is continuous on \( \mathbb{R}^2 \), we can switch the order of integration without affecting the outcome.

This theorem provides assurance that as long as the function is continuous over the domain, the integral's value remains unchanged regardless of whether we integrate x before y or y before x.

Continuous Functions

Continuous functions are those that have no interruptions, jumps, or breaks within their domain. This continuous nature is important for the applicability of Fubini's Theorem.

In the exercise problem, \( f \) is stipulated to be continuous on \( \mathbb{R}^2 \), which qualifies the use of Fubini's Theorem for interchanging orders of integration. Continuity ensures that for every point in the domain, small changes in input result in small changes in output.

This characteristic assures that the function maintains a smooth path over its domain, making integrations predictable and stable.

Order of Integration

The order of integration in double integrals refers to the sequence in which integrations occur between the variables. Unless otherwise specified, you will typically come across notations like \( \int \int f(x,y) \, dy \, dx \) or \( \int \int f(x,y) \, dx \, dy \), indicating the integrals of x and y done in sequence.

In mathematical problems like those in the exercise, sometimes it becomes advantageous to change the order of integration to simplify calculations. However, changing this order is only permissible under certain conditions, primarily the continuity of the function, as established by Fubini's Theorem.

By grasping the principle of the order of integration, especially via Fubini's Theorem, you can be more flexible and strategic in solving double integral problems efficiently.