Question

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

Short answer

The original integral was given by \(\int_{1}^{e} \int_{0}^{\ln x} f(x, y) dy dx\), and by reversing the order of integration, we obtain the new integral as \(\int_{0}^{1} \int_{e^y}^{e} f(x, y) dx dy\).

Step-by-step solution

Determine the region of integration

Given integral is \(\int_{1}^{e} \int_{0}^{\ln x} f(x, y) dy dx\). The bounds of the integral are: $$ 1 \le x \le e \\ 0 \le y \le \ln x $$ This can be rewritten as inequalities among x and y: $$ x \ge 1 \\ y \le \ln x $$ Based on the given bounds, the region of integration is the area in the xy-plane where x varies from 1 to e and y varies from 0 to ln(x).

Find the new limits of integration

To reverse the order of integration, we want to find the new limits of integration for x and y. We do this by expressing x in terms of y: $ x = e^y $ Now, we will determine the minimum and maximum values of x and y. From x = 1 to x = e, y can vary from 0 to 1 (because ln(1) = 0 and ln(e) = 1). Thus, we have the new limits of integration: $$ 0 \le y \le 1 \\ e^y \le x \le e $$

Rewrite the integral

Now that we have the new limits of integration, we can rewrite the integral with the new order of integration. The reversed integral is: $$\int_{0}^{1} \int_{e^y}^{e} f(x, y) dx dy$$

Key concepts

Double Integral

A double integral is a fancy term that essentially means we're integrating a function with two variables. In this case, these variables are usually written as \(x\) and \(y\). Imagine it like a sum, but instead of just summing over one line, we're summing over a whole area. This area is in the \(xy\)-plane.
When dealing with a double integral, it can be shown symbolically like this:

• \(\int \int f(x, y) \, dx \, dy\)

The function \(f(x, y)\) is the part we're integrating, and the symbols \(dx\) and \(dy\) indicate the areas where we're finding the integral. Double integrals help calculate volumes under surfaces. Now that we understand double integrals involve summing areas, let's see how those areas are defined.

Region of Integration

The region of integration is the area over which we calculate the double integral. Simply put, it tells you where you can evaluate the function.
Given a specified area on the xy-plane, you have conditions or boundaries on both \(x\) and \(y\) values. For example, in this exercise:

• \(1 \le x \le e\)
• \(0 \le y \le \ln x\)

This in essence gives us a "box" or "region" in which we are interested. This region is very crucial because it defines the limits - the start and end points - of our integration. Visualize it as fencing an area where all points (\(x, y\)) must lie within.
For the exercise, we found that the region is between \(x\) from 1 to \(e\) and \(y\) from 0 to \(\ln x\). Understanding this region helps in selecting correct integrals.

Limits of Integration

Limits of integration are the bounds that define the region in which the integration occurs. These bounds are where \(x\) and \(y\) start and stop.
In a double integral, we read from the outside integral to the inside. The exercise initially had:

• Outer integral for \(x\): \(1 \le x \le e\)
• Inner integral for \(y\): \(0 \le y \le \ln x\)

When we reverse the order of integration, these limits need re-evaluation. This requires expressing \(x\) in terms of \(y\). Here,

• \(y\) goes from 0 to 1 (\(0 \le y \le 1\))
• \(x\) varies between \(e^y\) and \(e\) (\(e^y \le x \le e\))

These newly adjusted limits allow us to integrate in the different order \(\int_{0}^{1} \int_{e^y}^{e} f(x, y) \, dx \, dy\). Setting the correct limits is necessary for solving the integral correctly.

Exponential Function

An exponential function is one where a constant base is raised to a variable exponent. Often, we see it represented as \(e^x\), where \(e\) is a special constant approximately equal to 2.71828.
In integrals involving these functions, such as our altered exercise, distinguishing these forms becomes crucial:

• \(y = \ln x\): The natural log, the inverse of an exponential function, defines one of our region’s boundaries.
• \(x = e^y\): Describes an exponential relationship where \(x\) is influenced by changes in \(y\).

This reversible property between an exponential and its logarithmic counterpart often simplifies integration processes. Understanding the exponential function is key in setting and re-evaluating the limits when swapping integration orders. Employing properties of exponential functions in integration helps in constructing and simplifying mathematical models in real-life applications.