Problem 37 - Calculus Early Transcendentals: Pearson New International Edition

Question

Write the given iterated integral as an iterated integral with the order of integration interchanged. Hint: Begin by sketching a region \(S\) and representing it in two ways, as in Example \(5 .\) $$ \int_{0}^{1} \int_{-y}^{y} f(x, y) d x d y $$

Step-by-Step Solution

Answer

The reordered integral is \( \int_{-1}^{1} \int_{0}^{|x|} f(x, y) \ dy \ dx \).

Step 1: Identify the given order of integration limits

The given iterated integral is \( \int_{0}^{1} \int_{-y}^{y} f(x, y) \ dx \ dy \). The limits for \(x\) are from \(-y\) to \(y\), and the limits for \(y\) are from \(0\) to \(1\).

Step 2: Sketch the region of integration

To identify the region \(S\), sketch the lines \(x = -y\), \(x = y\), \(y = 0\), and \(y = 1\). The intersection of these boundaries forms a triangular region in the xy-plane.

Step 3: Describe the region as a type II region

In the xy-plane, the region is bounded by \(y \geq 0\), \(y \leq 1\), \(x \geq -y\), and \(x \leq y\). This region is described in terms of \(y\) as a function of \(x\) for changing the order of integration.

Step 4: Rearrange the bounds for new order of integration

To integrate first with respect to \(y\), determine the limits based on the region. \(y\) ranges from \(-x\) to \(x\), while \(x\) ranges from \(0\) to \(1\).

Step 5: Write the iterated integral with interchanged order

The iterated integral with interchanged order is \( \int_{-1}^{1} \int_{0}^{|x|} f(x, y) \ dy \ dx \). The region is now defined with respect to \(x\).

Key Concepts

Order of Integration

In iterated integrals, the order of integration refers to the sequence in which the integral is taken with respect to each variable. For a double integral in the form \( \int \int f(x, y) \, dx \, dy \), it means integrating with respect to \(x\) first, followed by \(y\). Sometimes, it might be necessary to change the order of integration. This can simplify the problem or make the integration possible.
The original problem starts with integrating \(x\) from \(-y\) to \(y\), and then \(y\) from 0 to 1. After swapping, we integrate \(y\) first over \([-x, x]\) and then \(x\) from 0 to 1.

Region of Integration

The region of integration is the area over which the function is being integrated. It is crucial to first identify this region in the xy-plane, which helps us determine the limits for integration.
In the example, the specified region \(S\) is defined by:

• \(y \geq 0\)
• \(y \leq 1\)
• \(x \geq -y\)
• \(x \leq y\)

This forms a triangular shape in the xy-plane, bounded by the lines \(x = -y\), \(x = y\), \(y = 0\), and \(y = 1\). Understanding this region allows us to correctly change the order of integration.

Sketching Regions

Sketching the region of integration is a helpful visual tool that makes it easier to rearrange the bounds for integration. By plotting the boundaries \(x = -y\), \(x = y\), \(y = 0\), and \(y = 1\), you can clearly see the triangular region in the xy-plane. Sketching is essential because:

• It visually verifies the boundaries and their intersections.
• Helps determine new bounds when changing the order of integration.
• Ensures the region of integration is correctly represented.

Making an accurate sketch is a crucial step before attempting to change the order of integration in an iterated integral.

Integration Limits

Integration limits are the values that define the start and end points for the integral. These limits must be carefully analyzed, especially when the order of integration changes.
For the initial integral \(\int_{0}^{1} \int_{-y}^{y} f(x,y) \, dx \, dy\), the limits depend on the variables \(x\) and \(y\). After changing the order, the limits are identified as \( \int_{-1}^{1} \int_{0}^{|x|} f(x,y) \, dy \, dx\). Here:

• \(y\) is bounded by \([-x, x]\) when changing the integration order.
• \(x\) ranges from 0 to 1.

Correctly determining these limits ensures that the integration is performed over the correct region.