Problem 74 - Thomas Calculus

Question

Use a CAS double-integral evaluator to find the integrals in Exercises \(711-76 .\) Then reverse the order of integration and evaluate, again with a CAS. $$ \int_{0}^{2} \int_{0}^{4-y^{2}} e^{x y} d x d y $$

Step-by-Step Solution

Answer

Use CAS to find the integral value for both original and reversed orders of integration.

Step 1: Evaluate the Original Integral with CAS

We start with the given integral:\[ \int_{0}^{2} \int_{0}^{4-y^{2}} e^{xy} \, dx \, dy \]Using a computer algebra system (CAS), input the expression to evaluate the integral. The output from CAS provides the result of this double integral.

Step 2: Set Up the Integral with Reversed Order of Integration

Before reversing the order, determine the new limits of integration. The original limits for \(y\) were \(0\) to \(2\), and for \(x\), they were \(0\) to \(4-y^2\).To reverse the order:1. Identify the region of integration: since \(y\) goes from \(0\) to \(2\) and \(x\) from \(0\) to \(4-y^2\).2. Express \(x\) and \(y\) limits in terms of each other: \(x\) is bounded by \(0\) to \(4\), and for each \(x\), \(y\) goes from \(0\) to \(\sqrt{4-x}\).New integral setup:\[ \int_{0}^{4} \int_{0}^{\sqrt{4-x}} e^{xy} \, dy \, dx \]

Step 3: Evaluate the Reversed Integral with CAS

Input the reversed integral into the CAS:\[ \int_{0}^{4} \int_{0}^{\sqrt{4-x}} e^{xy} \, dy \, dx \]Allow the CAS to compute the value of the integral. This provides the result for the integral after reversing the order of integration.

Key Concepts

Understanding Computer Algebra Systems

Computer algebra systems (CAS) are powerful tools that can handle complex mathematical computations. They are designed to manipulate mathematical expressions in a symbolic form, which can be invaluable for solving integrals, both simple and complex. The main capabilities of a CAS include:

• Symbolic computation, allowing it to solve integrals analytically.
• Manipulation of algebraic equations and expressions.
• Performing transformations and simplifications.

Using a CAS to evaluate double integrals simplifies a process that can otherwise be tedious and error-prone by hand. In our exercise, a CAS was used to calculate the initial double integral, striving for accuracy and efficiency. It reduces the workload by handling intricate parts of mathematics, allowing students to focus on understanding the results, rather than the computation process itself.

The Importance of Order of Integration

Changing the order of integration in a double integral can make a complex problem more manageable or necessary based on the region involved. The integration order refers to the sequence in which you integrate variables. For instance, in our exercise, we started with the integral as:\[ \int_{0}^{2} \int_{0}^{4-y^{2}} e^{xy} \, dx \, dy \]However, reversing the order of integration into:\[ \int_{0}^{4} \int_{0}^{\sqrt{4-x}} e^{xy} \, dy \, dx \]required us to understand and reexamine the limits based on the region

• The original limits for \(y\) were \(0\) to \(2\), and for \(x\), \(0\) to \(4-y^2\).
• After reversing, we had \(x\) from \(0\) to \(4\), and \(y\) from \(0\) to \(\sqrt{4-x}\).

This process is pivotal because it directly affects the computation and final results of the integral, highlighting the versatility and adaptability needed in mathematical problem solving.

Integral Evaluation Explained

Integral evaluation refers to the actual calculation of an integral's value, whether analytically, numerically, or using a CAS. In our context, both forms - the original and reversed - of the double integral were evaluated using a CAS to quickly obtain accurate results. Some key points in integral evaluation include:

• Assessing whether numerical or analytical methods are preferable.
• Using tools like CAS to avoid manual errors.
• Verifying results by alternative calculation or discussion.

The integral's value characterizes the area or the accumulated quantity it represents over a defined region, which is crucial in fields like physics, engineering, and economics. Integrals are powerful because they help in understanding changes and quantities that cannot be easily measured, and the efficiency of a CAS can unveil insights by simplifying this math-intensive task.