Problem 72 - 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. $$ \int_{0}^{3} \int_{x^{2}}^{9} x \cos \left(y^{2}\right) d y d x $$

Step-by-Step Solution

Answer

The integral evaluates to the same result when the limits are reversed.

Step 1: Evaluate the Original Integral using CAS

To evaluate \( \int_{0}^{3} \int_{x^{2}}^{9} x \cos(y^{2}) \, dy \, dx \), use a computer algebra system (CAS). First, integrate with respect to \( y \) from \( y = x^2 \) to \( y = 9 \). This requires finding the antiderivative of \( x \cos(y^2) \) concerning \( y \). Then, execute the integration with respect to \( x \) from 0 to 3.

Step 2: Reversing the Order of Integration

Identify the region of integration in the \( xy \)-plane. Originally, for each \( x \) from 0 to 3, \( y \) ranges from \( x^2 \) to 9. Draw the region bounded by \( y = x^2 \), \( y = 9 \), and \( x = 3 \). To reverse, consider \( y \) instead going from \( 0 \) to \( 9 \) and for each \( y \), \( x \) varies from 0 to \( \sqrt{y} \).

Step 3: Set Up the Reversed Integral

Write the integral with the reversed order: \( \int_{0}^{9} \int_{0}^{\sqrt{y}} x \cos(y^2) \, dx \, dy \). This integral accounts for \( x \) going from 0 to \( \sqrt{y} \) and \( y \) from 0 to 9.

Step 4: Evaluate the Reversed Integral using CAS

Use a CAS to evaluate the new integral \( \int_{0}^{9} \int_{0}^{\sqrt{y}} x \cos(y^2) \, dx \, dy \). First, find the integral concerning \( x \) from 0 to \( \sqrt{y} \), and then integrate the resultant expression with respect to \( y \) from 0 to 9.

Key Concepts

Order of Integration

When dealing with double integrals, the order of integration is significant. The order determines which variable you integrate with first. In the original exercise, we started by integrating with respect to \( y \) first and then \( x \), denoted as \( \int_{0}^{3} \int_{x^2}^{9} \). This means for each fixed \( x \) value, \( y \) ranges from \( x^2 \) to 9.
However, sometimes it's beneficial to reverse the order of integration. This involves changing which variable you integrate with first. It often simplifies the integration process or makes computation possible if a direct evaluation is difficult. By reversing the order, the variable \( y \) is integrated first from 0 to 9, and \( x \) is based on \( y \), spanning from 0 to \( \sqrt{y} \).
To successfully reverse the order, visualize the region of integration in the plane. This ensures each variable's bounds are correctly identified and rearranged.

Computer Algebra System

A Computer Algebra System (CAS) is a powerful tool used to perform complex mathematical computations symbolically rather than numerically. When dealing with intricate integrals, such as those involving non-trivial antiderivatives or multiple variables, a CAS can streamline your work.

• First, a CAS helps by automating the computation of antiderivatives, handling cumbersome algebraic expressions without error.
• Second, it performs evaluations of these expressions quickly, especially when reversing orders or when exact answers are needed.
• Lastly, once set up, it offers visualizations or verification steps, ensuring the validity of the computed results.

Using a CAS becomes especially useful when integrating functions that involve trigonometric expressions or unusual limits. It assures that you attain the correct results by providing consistent and efficient computation capabilities.

Antiderivative

An antiderivative is a function that reverses the process of differentiation. For each function, finding its antiderivative is vital in integration, as it allows you to evaluate definite integrals.
In our case, while integrating \( x\cos(y^2) \), you begin by finding an antiderivative concerning \( y \). This can be complex due to the presence of \( y^2 \) within a trigonometric function.

• Trigonometric antiderivatives, like those involving \( \cos \), often require substitution or integration by parts techniques.
• Identifying an appropriate antiderivative is necessary before performing any integration over set limits.
• When reversing integration order, antiderivatives must be recalibrated for the new variable bounds.

Finding the correct antiderivative is crucial as it ensures that the integration yields accurate results. Given the complexity of some functions, relying on a CAS can assist in this step to prevent errors and simplify calculations.