Question

Determine whether the following statements are true and give an explanation or counterexample. a. In the iterated integral \(\int_{c}^{d} \int_{a}^{b} f(x, y) d x d y,\) the limits \(a\) and \(b\) must be constants or functions of \(x\). b. In the iterated integral \(\int_{c}^{d} \int_{a}^{b} f(x, y) d x d y,\) the limits \(c\) and \(d\) must be functions of \(y\). c. Changing the order of integration gives \(\int_{0}^{2} \int_{1}^{y} f(x, y) d x d y=\int_{1}^{y} \int_{0}^{2} f(x, y) d y d x\).

Short answer

Overall, all three statements are incorrect: a) The limits a and b can be constants or functions of the other variable (y), not necessarily related to the range of x. b) The limits c and d can be constants or functions of the other variable (x), not necessarily related to the range of y. c) Changing the order of integration involves analyzing the range of each variable and adjusting the limits accordingly, rather than simply switching the limits.

Step-by-step solution

Statement a: Limits a and b

In an iterated integral of the form \(\int_{c}^{d} \int_{a}^{b} f(x, y) d x d y\), the limits \(a\) and \(b\) are related to the range of values the variable \(x\) can take. These limits can be constants or they can be functions of the other variable, \(y\) (not \(x\)) depending on the region we are integrating over. Therefore, the statement is incorrect as written.

Counterexample for Statement a

Consider the region defined by \(0 \le y \le 2\) and \(0 \le x \le y\). The iterated integral of a function \(f(x,y)\) over the region is given by: $$ \int_{0}^{2} \int_{0}^{y} f(x, y) d x d y $$ In this case, the limits \(a = 0\) and \(b = y\) depend on \(y\), not \(x\).

Statement b: Limits c and d

In an iterated integral of the form \(\int_{c}^{d} \int_{a}^{b} f(x, y) d x d y\), the limits \(c\) and \(d\) are related to the range of values the variable \(y\) can take. These limits can be constants or they can be functions of the other variable, \(x\). Therefore, the statement is incorrect as written.

Counterexample for Statement b

Consider the region defined by \(0 \le x \le 2\) and \(x \le y \le 2\). The iterated integral of a function \(f(x,y)\) over the region is given by: $$ \int_{0}^{2} \int_{x}^{2} f(x, y) d y d x $$ In this case, the limits \(c = x\) and \(d = 2\) do not depend on \(y\), which contradicts the statement.

Statement c: Changing the order of integration

The given iterated integral is: $$ \int_{0}^{2} \int_{1}^{y} f(x, y) d x d y $$ Changing the order of integration involves finding the new limits that represent the same region of integration. The given statement incorrectly suggests that we can simply switch the limits, but this is not the case. We need to analyze the range of values for each variable and adjust the limits accordingly.

Counterexample for Statement c

We can visualize the original integral's region as having \(x\) vary from \(1\) to \(y\), and \(y\) varying from \(0\) to \(2\). To switch the order of integration, \(y\) must be expressed in terms of \(x\). In this region, \(y\) varies from \(x\) to \(2\), and \(x\) varies from \(1\) to \(2\). The correct change of order would be: $$ \int_{0}^{2} \int_{1}^{y} f(x, y) dx dy = \int_{1}^{2} \int_{x}^{2} f(x, y) dy dx $$ This counterexample shows that the given statement c is incorrect.

Key concepts

Double Integral

Double integrals extend the concept of a single integral to two dimensions, allowing us to calculate the volume under a surface over a certain region. When we set up a double integral, we write it as \( \int\int f(x, y) dx dy \) with \( f(x, y) \) being the function we're integrating, and \( dx \) and \( dy \) indicating that we're integrating with respect to \( x \) and \( y \) over a region in the \( xy \) plane.

The limits of integration for both variables are crucial as they define the region over which we're calculating the area or volume. These limits can be constant values or functions that depend on the other variable. For example, for \( \int_a^b \int_c^d f(x, y) dx dy \) if \( a \) and \( b \) are constants, we're integrating over a rectangle. However, if either is a function of \( y \)—say \( a(y) \) and \( b(y) \)—the region of integration could be more complex, such as a region bounded by curves.

When solving double integrals, it's essential to understand the geometry of the region you're integrating over. Visualizing or sketching the region can often make the process much clearer and can prevent errors when setting up the integral.

Order of Integration

The order of integration refers to the sequence in which we integrate a multivariable function in a multiple integral. In a double integral, this would mean choosing whether to integrate with respect to \( x \) first (\( dx \) inside) and then \( y \) (\( dy \) outside), or the other way around. The order of integration can be vital in simplifying the computation, as sometimes one order may lead to an easier or even feasible integral when the other does not.

When dealing with integrals over more complex regions, one order of integration may be more natural based on the limits. For instance, if the limits for \( x \) are functions of \( y \) (as seen in the counterexample for statement a), it might be simpler to integrate with respect to \( x \) first. Conversely, if the \( y \) limits are functions of \( x \) (as in the counterexample for statement b), it might be easier to integrate with respect to \( y \) first.

Changing the order of integration does not change the value of the integral if done correctly. It requires careful consideration of the region of integration to ensure that the new limits accurately represent the same area and that no parts of the region are omitted or counted twice.

Integration Limits

Integration limits define the range over which we are integrating. In the context of double integrals, these limits are crucial since they delineate the region in the plane where the function \( f(x, y) \) is defined and where we are calculating the area or volume.

For a given iterated integral, the integration limits can be constant or they can vary, being expressed as functions of the other variable. In the exercise, statements suggesting that limits must be constants or functions of the same variable they are integrating with are incorrect. Instead, as demonstrated in the solutions, \( a \) and \( b \) can be functions of \( y \) and \( c \) and \( d \) can be functions of \( x \) based on the geometry of the integration region.

It's important not to assume that the limits can simply be switched when changing the order of integration. A new analysis of the integration domain is required to ensure the proper limits are determined for the new integral. This analysis avoids errors and ensures the resulting integral correctly represents the desired region, as seen in the counterexample for statement c.