Question

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

Question: Combine the given double integrals into a single iterated integral. Graph and describe the region of integration. Integral 1: $$\int_{-1}^{0} \int_{e^y}^{e} f(x, y) dx dy$$ Integral 2: $$\int_{0}^{1} \int_{e^{-y}}^{e} f(x,y) dx dy$$ Answer: The combined iterated integral is $$\int_{-1}^{1} \int_{e^{-|y|}}^{e} f(x, y) dx dy$$, and the region of integration includes both the first and third quadrants bounded by the curves \(x=e^y\), \(x=e^{-y}\), and the vertical line at \(x=e\).

Step-by-step solution

Graph the regions of integration

First, we graph the given boundaries for our region of integration. We have the following: 1. \(y = 0\), which is the line along the x-axis 2. \(x = e^y\), this is an exponential function of the variable \(y\) 3. \(x = e^{-y}\), this is an exponential function of the opposite of the variable \(y\) 4. \(x = e\), a vertical line at the value \(x=e\) The regions of integration for both integrals lie within the bounds of these curves.

Define the regions of integration

The limits of the integrals tell us the regions of integration. The first integral corresponds to the region in the first quadrant between the exponential function \(x=e^y\), the \(y=0\) line (i.e., the x-axis), and the vertical line at \(x=e\). The second integral corresponds to the region bounded by the exponential function \(x=e^{-y}\), the \(y=0\) line, and the vertical line at \(x=e\) in the third quadrant (below the x-axis).

Combine the integrals

Since the regions of integration for both integrals are adjacent, we can combine them into a single iterated integral. The region of integration will now include both the first and third quadrants bounded by the curves \(x=e^y\), \(x=e^{-y}\), and the vertical line at \(x=e\). So, to write the integrals as a single iterated integral, we will have: $$\int_{-1}^{1} \int_{e^{-|y|}}^{e} f(x, y) dx dy$$ Here, we used the absolute value function to account for both positive and negative values of \(y\).

Key concepts

Iterated Integrals

Understanding iterated integrals is crucial for students tackling complex calculus problems. An iterated integral involves performing multiple integrations, one inside another. When we look at the given exercise, it requires us to merge two separate integrals into a single iterated integral. This process streamlines the calculation, allowing us to consider the entire area of interest at once, instead of segmenting it into smaller parts.

The key to combining integrals is recognizing when different sections of a graph can be described using one encompassing range of integration. For example, in our exercise, the regions in the first and third quadrants can be combined because they share common boundaries and together form a complete section of the graph that is symmetrical about the y-axis. This approach highlights the power and elegance of iterated integrals in simplifying the integration process over complex regions.

Exponential Functions

Exponential functions such as \( e^y \) and \( e^{-y} \) play a pivotal role in calculus, especially when it comes to integration. These functions are unique as they are their own derivatives, which makes them essential in the study of growth and decay. In the exercise, we encounter two exponential functions that define the bounds of the region we are integrating over.

When graphing exponential functions, it's important to remember that \( e^y \) grows rapidly as y increases, while \( e^{-y} \) does the opposite and approaches zero. This behavior affects the shape of the region we are trying to integrate. Recognizing the characteristics of these functions helps us visualize the area under the curve and better understand how to set up our integrals accordingly.

Graphing Integration Bounds

Graphing integration bounds is an invaluable skill for visualizing the area over which you're integrating. In our exercise, the bounds are defined by both linear and exponential equations. Graphing these correctly allows us to see the exact region we're interested in.

By drawing vertical and horizontal lines corresponding to constant values, and curved lines representing the exponential functions, we map out the space on the coordinate plane where our function exists. This visual aid is immensely helpful when it becomes necessary to merge the regions under different bounds, as in the case of this exercise. Through graphing, we see how the regions connect, justifying the use of absolute values to consolidate the bounds for a single iterated integral.

Absolute Value in Integration

The use of absolute values in integration is a clever mathematical tool that helps us deal with symmetry in our region of integration. As shown in the combined iterated integral for our exercise, \( e^{-|y|} \) accounts for both the positive and negative values of y. This integral effectively covers both the first and third quadrants with one expression.

The absolute value ensures that the integration bounds are positive regardless of y's sign, reflecting the symmetry of the exponential function on either side of the y-axis. This approach not only simplifies the integral but also streamlines the calculation process, providing a clear path to follow for the evaluation of the integral over these symmetrical regions.