Question

Two integrals to one Draw the regions of integration and write the following integrals as a single iterated integral: $$\int_{0}^{1} \int_{e^{y}}^{e} f(x, y) d x d y+\int_{-1}^{0} \int_{e^{-y}}^{e} f(x, y) d x d y$$

Short answer

Based on the given step by step solution, consider the following short answer: "To combine the two given integrals into a single iterated integral, we first analyze the individual integration regions and find their intersection. We then determine new limits for y, which vary from -1 to 1. For x, we use a piecewise function for the lower bound depending on the range of y: \(e^{-y}\) for y in [-1, 0) and \(e^{y}\) for y in [0, 1]. The combined integral is written as: \(\int_{-1}^{1} \int_{e^{-\lvert y \rvert}}^{e} f(x, y) dx dy.\)"

Step-by-step solution

Understand the individual integration regions

Let's first look at the individual integration regions for both given integrals. For the first integral, $$\int_{0}^{1} \int_{e^{y}}^{e} f(x, y) dx dy,$$ y varies from 0 to 1, and x varies from \(e^y\) to e. This region describes a curve enclosed by the lines y = 0, y = 1, x = \(e^y\), and x = e. For the second integral, $$\int_{-1}^{0} \int_{e^{-y}}^{e} f(x, y) dx dy,$$ y varies from -1 to 0, and x varies from \(e^{-y}\) to e. This region describes a curve enclosed by the lines y = -1, y = 0, x = \(e^{-y}\), and x = e.

Find the combined region

Now, we will find the combined region of these two individual integration regions. Observe that both regions have the same upper bound for x (x = e), so we don't need to modify these bounds. To combine both regions, we need to extend the limits for y to cover both ranges. The new limits for y will vary from -1 to 1. Next, we notice that for y in [-1, 0), x varies from \(e^{-y}\) to e, and for y in [0, 1], x varies from \(e^y\) to e. To write these bounds, we can use a piecewise function for the lower bound of x. Therefore, the combined region for both integrals can be expressed with the following bounds: $$ \int_{-1}^{1} f(x, y) dx dy \qquad \text{where} \\ x \text{ varies from } \begin{cases} e^{-y} & \text{if } -1 \leq y < 0, \\ e^{y} & \text{if } 0 \leq y \leq 1, \end{cases} \\ \text{and} \quad y \text{ varies from } -1 \text{ to } 1. $$

Write the single iterated integral

With the combined region determined, we can now combine the given integrals into a single iterated integral. Let's use the bounds that we found in Step 2: $$\int_{-1}^{1} \int_{e^{-\lvert y \rvert}}^{e} f(x, y) dx dy.$$ And that's it! We have now successfully combined the two integrals into a single iterated integral.

Key concepts

Region of Integration

A region of integration defines the specific area over which an integral is calculated. When dealing with double integrals, the region is a two-dimensional area in the coordinate plane. For the given problem, each of the integrals involves a different region of integration.
Consider the first integral, where the region of integration is described by:

• The variable \(y\) ranges between 0 to 1.
• The variable \(x\) ranges between \(e^y\) and \(e\).

This forms a region bounded vertically between \(y=0\) and \(y=1\), and horizontally between the curves \(x=e^y\) and \(x=e\).
For the second integral:

• \(y\) ranges between -1 to 0.
• \(x\) ranges between \(e^{-y}\) and \(e\).

This creates a region bounded similarly, but on the negative part of the \(y\)-axis. To combine these regions into a single integration area, adjust \(y\) to range from -1 to 1 while accommodating changing equations for \(x\).

Piecewise Function

A piecewise function defines different expressions for different parts of its input range. This technique is essential when combining integrals over regions with distinct behaviors or boundary conditions, just like the current problem.
In our problem, the lower bounds of \(x\) vary depending on the value of \(y\). To handle this:

• If \(y\) is between -1 and 0, we use \(x = e^{-y}\)
• If \(y\) is between 0 and 1, we use \(x = e^y\)

This results in a piecewise function that matches the characteristics of both original integrals in one unified expression. By using a piecewise function, you can effectively account for the distinct bounds from each section of the region of integration.

Bounds of Integration

The bounds of integration specify the limits between which the integration is performed. They define where the function begins and ends along each axis.

• Vertical limits: These are determined by the \(y\)-values which range from -1 to 1.
• Horizontal limits: The name and form are identified by a piecewise function where \(x\) is integrated. This ranges from either \(e^{-y}\) or \(e^y\) to \(e\) depending on the \(y\)-value.

Bounds are crucial as they directly influence the integrand's area or volume. For the unified integral \(\int_{-1}^{1} \int_{e^{-\lvert y \rvert}}^{e} f(x, y) \, dx \, dy\), we use the function \(e^{-\lvert y \rvert}\) to capture the lower bound, seamlessly combining the behaviors for both the negative and positive \(y\) regions previously treated independently. The area enclosed within these bounds represents the final region over which the integration is executed.