Question

Reverse the order of integration in the following integrals. $$\int_{0}^{1} \int_{1}^{e^{y}} f(x, y) d x d y$$

Short answer

Question: Reverse the order of integration for the given integral. Integral: $$\int_{0}^{1} \int_{1}^{e^{y}} f(x, y) d x d y$$ Answer: After reversing the order of integration, the new integral is: $$\int_{1}^{e} \int_{\ln{x}}^{1} f(x, y) d y d x$$.

Step-by-step solution

Understand the current order of integration and limits

The integral is given by: $$\int_{0}^{1} \int_{1}^{e^{y}} f(x, y) d x d y$$ Here, the outer integral is with respect to y and has limits 0 to 1, while the inner integral is with respect to x and has limits 1 to \(e^y\).

Draw a region of integration

Plot the limits of integration and determine the area of integration. The region is bounded by y = 0, y = 1, x = 1, and x = \(e^y\). This region is defined by an area in the xy-plane.

Swap the order of integration and find the new limits

To reverse the order of integration, we will integrate first with respect to y and then with respect to x. To find the new limits, we need to express y in terms of x. From x = \(e^y\), we have y = ln(x). Thus, the new lower limit for y is ln(x) and the upper limit is 1 since it does not change. Then, the x-limits remain 1 and \(e\), the maximum value of x. Our reversed integral will be: $$\int_{1}^{e} \int_{\ln{x}}^{1} f(x, y) d y d x$$

Write the final reversed order of integration integral

The reversed order of integration for the given integral is: $$\int_{1}^{e} \int_{\ln{x}}^{1} f(x, y) d y d x$$

Key concepts

Order of Integration

In the world of double integrals, reversing the order of integration is a frequent and essential technique. Initially, the given integral has the order of integration as \( \int_{0}^{1} \int_{1}^{e^{y}} f(x, y) \, dx \, dy \). Here, integration occurs first with respect to \(x\), given the inner integral \( \int_{1}^{e^{y}} \, dx \), followed by \(y\) in the outer integral \( \int_{0}^{1} \, dy \). For complex regions, reversing the order can streamline computation.

• The initial boundaries are with \(x\) ranging from 1 to \(e^y\) and \(y\) from 0 to 1.
• Reversing it entails integrating \(y\)-first, altering our computational strategy.

First, identify the region of integration visually. Once the region is understood, the new order can simplify solving. With the swapped order, \( y \) adopts limits \( \ln(x) \) to 1, while \(x\) expands from 1 to \(e\). Thus, understand both orders to confidently switch them where conducive.

Limits of Integration

The limits of integration are the defining boundaries of the integral, crucial for determining over which region the function \(f(x, y)\) is integrated. Originally, for the given integral, the limits for \(x\) were 1 to \(e^y\) and for \(y\), from 0 to 1. These boundaries outlined a specific area in the xy-plane.

Understanding the region in the plane helps in visualizing and setting up double integrals. Upon reversing the order, understanding these limits is vital as they will change to reflect the new orientation of integration. After reversing, the new setup involves:

• \( x \) having static limits from 1 to \( e \).
• \( y \) adapting to dynamic limits \( \ln(x) \) to 1, as defined earlier.

Visual exercises are beneficial in properly managing these transitions, ensuring the new setup accurately represents the same region initially described. Altering the setup without manipulating these correctly changes the integral completely. Therefore, calculate with precision.

Integration Techniques

Integration is a fundamental concept in calculus involving techniques that simplify complex problems. In the context of double integrals, several methods can be employed to evaluate or simplify them, especially when changing the order of integration.

Reversing order or changing bounds can transform intimidating integrals into attainable problems. Here are some techniques to keep in mind:

• **Substitution**: Useful when direct integration is challenging, allowing variables to change making functions simpler.
• **Partial Fractions**: This technique breaks down complex fractions into simpler pieces, often used with polynomial denominators.
• **Table of Integrals**: A handy reference of common integral forms, easing the workload by consulting proven results.

Being familiar with these techniques not only aids in the computation of straightforward integrals but also in the rearrangement and solving of double integrals. The experience with a variety of problems strengthens the understanding of when and how to apply these tools efficiently. With practice, mastering integration techniques leads to strategic decision-making on how best to approach any given integral in calculus.