Problem 85 - Thomas Calculus

Question

Use a CAS double-integral evaluator to estimate the values of the integrals in Exercises \(85-88 .\) \begin{equation}\int_{1}^{3} \int_{1}^{x} \frac{1}{x y} d y d x\end{equation}

Step-by-Step Solution

Answer

The estimated value of the integral is approximately 0.603.

Step 1: Understand the Integral Setup

The given double integral is \( \int_{1}^{3} \int_{1}^{x} \frac{1}{x y} \, dy \, dx \). This means we are integrating the function \( \frac{1}{xy} \) over a region defined by \( 1 \leq y \leq x \) for each \( x \) where \( x \) ranges from 1 to 3.

Step 2: Inner Integral Evaluation

First, integrate \( \frac{1}{xy} \) with respect to \( y \) from 1 to \( x \). This inner integral becomes \( \int_{1}^{x} \frac{1}{xy} \, dy \). It is equal to \( \frac{1}{x} \int_{1}^{x} \frac{1}{y} \, dy \), which simplifies to \( \frac{1}{x} [\ln|y|]_{1}^{x} = \frac{1}{x} [\ln(x) - \ln(1)] = \frac{1}{x} \ln(x) \).

Step 3: Outer Integral Evaluation

Next, integrate the result \( \frac{1}{x} \ln(x) \) with respect to \( x \) from 1 to 3. This outer integral is \( \int_{1}^{3} \ln(x) \frac{1}{x} \, dx \). Notice that \( \ln(x) \frac{1}{x} \) is the derivative of \( \frac{(\ln(x))^2}{2} \). Thus, the integral computes to \( \left[ \frac{(\ln(x))^2}{2} \right]_{1}^{3} \).

Step 4: Compute the Definite Integral

Evaluate \( \frac{(\ln(x))^2}{2} \) from 1 to 3: \( \left( \frac{(\ln(3))^2}{2} \right) - \left( \frac{(\ln(1))^2}{2} \right) = \frac{(\ln(3))^2}{2} - 0 = \frac{(\ln(3))^2}{2} \).

Step 5: Final Calculation with CAS

Using a Computer Algebra System (CAS), evaluate \( \frac{(\ln(3))^2}{2} \) numerically to get an approximate value. If calculated, \( \ln(3) \approx 1.0986 \), leading to \( \left(\frac{(1.0986)^2}{2}\right) \approx 0.603 \).

Key Concepts

Iterated Integrals

Iterated integrals are a fundamental concept when dealing with multiple variable calculus, particularly in double integration. Double integrals allow you to compute the volume under a surface across a defined region, and iterated integrals help break this down step by step.
To understand this, let's consider the given exercise where the function \(\frac{1}{xy}\) is integrated over a region.The process of handling iterated integrals involves integrating one variable at a time.

• You start with the innermost integral for the variable that has the fixed limits based on the outermost variable's value.
• In this case, the inner integral \(\int_{1}^{x} \frac{1}{xy} \, dy\) simplifies to \(\frac{1}{x} \ln(x)\).

Once the inner integration is complete, the obtained function becomes the subject of the outer integral. Following this step-by-step process helps efficiently manage complex integrations and understand the transformation of function forms.
This breakdown makes it easier to comprehend how variables interdependently change within the defined limits.

Definite Integrals

Definite integrals, unlike indefinite integrals, involve bounds that specify the limits of integration. They allow us to find the exact area under a curve between two specified points. In the case of double integrals, they help define the limits for both variables across which the integration is conducted.
For the given problem, the bounds of integration are \(1 \leq y \leq x\) and \(1 \leq x \leq 3\). These bounds precisely state where the evaluation of the integral needs beginning and ending.The specific calculations involve taking the antiderivative of the function,

• Beginning with the innermost integration, and then working towards the outer limits, yields the desired outcome, \( \frac{(\ln(3))^2}{2}\).

Ultimately, a definite integral not only gives the area but simplifies to provide a numeric value, as seen when using a Computer Algebra System (CAS). This numeric evaluation offers an efficient way of understanding the total sum of areas.

Computer Algebra System (CAS)

A Computer Algebra System, or CAS, fundamentally enhances how we perform complex mathematical expressions, particularly integrals.
A CAS can automate the calculation process, manage algebraic expressions effortlessly, and provide numerical approximations with great precision. This capability was highlighted in the step where \(\frac{(\ln(3))^2}{2}\) needed to be evaluated.The benefits of CAS include:

• Quickly solving intricate problems that would be tedious by hand.
• Delivering results with more accuracy, which are crucial for advanced mathematical operations.

In educational settings, a CAS is invaluable, as it helps students verify results, explore complex integrations, and understand underlying mathematical concepts better without getting bogged down with repetitive arithmetic. For instance, calculating \(\ln(3)\) and then squaring it, followed by evaluating with integrated bounds is simplified using a CAS.
This not only saves time but helps solidify computational understanding through instant feedback.