Question

In Exercises \(59-62,\) use a computer algebra system to approximate the iterated integral. $$ \int_{0}^{2} \int_{0}^{4-x^{2}} e^{x y} d y d x $$

Short answer

Unfortunately without a specific CAS system, a numerical approximation can't be provided. Following the steps will yield a numerical result within the CAS system.

Step-by-step solution

Inner Integral

Firstly, perform the integration with respect to \(y\), which is the inner integral \( \int_{0}^{4-x^{2}} e^{x y} d y\). This results in the integral \(\frac{e^{xy}}{x}\) but evaluated from \(y = 0\) to \(y = 4 - x^2\), which ends up being \( \frac{e^{4x - x^3}}{x} - \frac{1}{x}\).

Outer Integral

Then, the result from Step 1 is inserted into the outer integral \(\int_{0}^{2}\), resulting in \(\int_{0}^{2}\left(\frac{e^{4x - x^3}}{x} - \frac{1}{x}\right) dx\).

Approximation using a computer algebra system

The last step is to use a computer algebra system like Mathematica, Maple or similarly, to evaluate the integral from step 2. Depending on the software used the command could look different, but the concept stays the same.

Key concepts

Computer Algebra System

Computer Algebra Systems (CAS) are powerful tools used to perform complex mathematical operations, such as simplifications, factorizations, solving equations, and most relevant to this exercise, integrating functions. They are essential when it comes to integrals that are difficult or impossible to compute by hand, like the iterated integral in our exercise.

A CAS can save time and improve accuracy by handling the computational complexity of the task. For students, it is a valuable resource when verifying their work or when they need to approximate values for non-exact integrals. However, while a CAS provides answers quickly, it's important to understand the underlying mathematics to use these tools effectively. When working with CAS, always remember to input your functions accurately and know the correct commands for your specific system, whether it's Mathematica, Maple, or another software.

Double Integration

Double integration is a process used to compute the volume under a surface in a two-dimensional space. The concept extends from single integration, which finds the area under a curve. When dealing with double integrals, the function is typically integrated with respect to one variable first, called the inner integral, then integrated with respect to the second variable, known as the outer integral.

To compute our exercise's iterated integral, we first hold x constant and integrate with respect to y, then integrate the result with respect to x. It's vital to pay attention to the limits of integration for each variable, which may depend on the other variable, as in the case of the upper limit for y in our exercise, which is expressed as \(4-x^2\). During this process, understanding the regions of integration and the behavior of the function within these regions can further contribute to a successful computation.

Exponential Functions

Exponential functions are fundamental in various areas of mathematics and appear frequently in calculus problems, including the one from our textbook exercise. They are defined as functions of the form \(f(x) = a^x\), where the base \(a\) is a constant, and \(x\) is the variable. The most common base is Euler's number, \(e\), which is approximately 2.71828.

These functions are characterized by their rapid growth or decay and have unique properties that make them quite interesting to work with. For example, the derivative and integral of the natural exponential function \(e^{x}\) are the same as the function itself. In our context, when integrating exponential functions like \(e^{xy}\), it's important to consider the presence of other variables and apply the proper integration techniques or rules. Even with a CAS at one's disposal, understanding these functions' behavior, such as how their rate of change affects area and volume calculations, is invaluable.

Incorporating exponential functions into a double integration, as shown in the textbook exercise, requires students to not just process the computation but also understand the interplay between the exponential components and the changing limits of integration.