Problem 38 - University Calculus: Early Transcendentals

Question

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$\int_{0}^{\ln 2} \int_{e^{y}}^{2} d x d y$$

Step-by-Step Solution

Answer

The reversed integral is \( \int_{1}^{2} \int_{0}^{\ln x} dy \, dx \).

Step 1: Identify the Region of Integration

First, understand the limits of integration. The given integral is \( \int_{0}^{\ln 2} \int_{e^{y}}^{2} dx \, dy \). This means that for a fixed \( y \), \( x \) varies from \( e^y \) to \( 2 \). Similarly, \( y \) varies from \( 0 \) to \( \ln 2 \).

Step 2: Sketch the Region

On the \( xy \)-plane, plot the lines \( x = e^y \) and \( x = 2 \). The region of integration is bounded on the left by the curve \( x = e^y \), on the right by the vertical line \( x = 2 \), on the bottom by \( y = 0 \), and on the top by \( y = \ln 2 \). This forms a region that is bounded by the curve \( x = e^y \) until it intersects with \( y = \ln 2 \), and then bounded by the vertical line \( x = 2 \).

Step 3: Determine the Bounds for y in Terms of x

To reverse the order of integration, express \( y \) in terms of \( x \). Since, \( x = e^y \), \( y \, = \, \ln x \). Within the defined region, \( x \) ranges from \( 1 \) (since \( e^0 = 1 \)) up to \( 2 \). Correspondingly, \( y \) ranges from \( 0 \) (when \( x = 1 \)) to \( \ln x \).

Step 4: Express the Reversed Double Integral

With the new bounds established, the reversed double integral is \( \int_{1}^{2} \int_{0}^{\ln x} dy \, dx \). This integral represents the same region but integrates first with respect to \( y \) from \( 0 \) to \( \ln x \) for a given \( x \), and then with respect to \( x \) from \( 1 \) to \( 2 \).

Key Concepts

Order of Integration

The order of integration in a double integral is pivotal as it dictates the sequence in which we integrate with respect to each variable. In the context of the original exercise, the given order is first integrating with respect to \( x \) and then \( y \), as seen in the integral format \( \int_{0}^{\ln 2} \int_{e^{y}}^{2} dx \, dy \). This means for each slice of \( y \), you perform the integration over \( x \).
When reversing the order of integration, the roles of \( x \) and \( y \) are swapped, which sometimes simplifies calculations or is necessary due to the nature of the functions involved. After reversing, our new order integrates \( y \) first, followed by \( x \): \( \int_{1}^{2} \int_{0}^{\ln x} dy \, dx \).

• Integrating with respect to \( x \) first requires resolving \( x \) boundaries within each layer of constant \( y \).
• Integrating with respect to \( y \) first necessitates resolving \( y \) boundaries for each \( x \).

Understanding the order of integration allows us to manipulate integrals to better understand or solve a given problem.

Region of Integration

In double integrals, the region of integration refers to the domain over which the integrals are being evaluated. This region is defined by the limitations in the bounds of integration.
For the integral problem \( \int_{0}^{\ln 2} \int_{e^{y}}^{2} dx \, dy \), the region is determined by \( x \) values ranging between \( e^y \) to \( 2 \) and \( y \) values between \( 0 \) to \( \ln 2 \).
Visualizing this in the \( xy \)-plane involves:

• Bounding the left by the curve \( x = e^y \).
• Bounding the right by the vertical line \( x = 2 \).
• Confined between \( y = 0 \) (bottom) and \( y = \ln 2 \) (top).

Upon sketching, the region forms a typical shape often described in these kinds of problems. It is necessary to correctly interpret these boundaries to set up the reversed order of the integral accurately.

Integration Bounds

Determining the integration bounds is a crucial step in setting up both single and double integrals. The bounds tell you the range of values for each variable and are necessary for accurate calculations.
In our original integral expression, \( x \) varies between \( e^y \) and \( 2 \), while \( y \) varies from \( 0 \) to \( \ln 2 \). These bounds set the limits of integration for both variables under the original order.
After changing the order of integration, the relationship and bounds require adjustment. We express \( y \) in terms of \( x \), where \( y = \ln x \). Then, the integration limits become:

• For \( y \), range from \( 0 \) to \( \ln x \).
• For \( x \), range from \( 1 \) to \( 2 \) (notably because \( e^0 = 1 \)).

These bounds are integral in redefining the double integral to maintain the integrity of the region's exposure from the original problem.