Problem 42 - Calculus Early Transcendentals: Pearson New International Edition

Question

Approximate \(\int_{0}^{a} \frac{1}{\pi}\left(1+x^{2}\right)^{-1} d x\) for \(a=10,50\), and \(100 .\)

Step-by-Step Solution

Answer

Approximate integrals: for 10, 0.4680; for 50, 0.4936; for 100, 0.4971.

Step 1: Identify the Function and Constants

We need to approximate the integral \( \int_{0}^{a} \frac{1}{\pi} \left(1+x^{2}\right)^{-1} dx \). The constant \( \frac{1}{\pi} \) can be factored out of the integral, which simplifies our computations.

Step 2: Recognize the Function's Formula

The function \((1+x^2)^{-1}\) is known as the arctangent function's derivative. This means the indefinite integral of \((1+x^2)^{-1}\) is \(\tan^{-1}(x)\).

Step 3: Compute the Indefinite Integral

Calculate \(\int (1+x^2)^{-1} \, dx = \tan^{-1}(x) + C\), where \(C\) is a constant of integration.

Step 4: Apply Definite Integral Limits

The definite integral from 0 to \(a\) becomes \[\int_{0}^{a} (1+x^2)^{-1} \, dx = \tan^{-1}(a) - \tan^{-1}(0) = \tan^{-1}(a)\] since \(\tan^{-1}(0) = 0\).

Step 5: Calculate for Specific Values of \(a\)

1. For \(a = 10\), the integral is \[\tan^{-1}(10) \approx 1.4711\] radians.2. For \(a = 50\), the integral is \[\tan^{-1}(50) \approx 1.5508\] radians.3. For \(a = 100\), the integral is \[\tan^{-1}(100) \approx 1.5608\] radians.

Step 6: Multiply by the Constant Factor

Since the integral is originally \( \frac{1}{\pi} \times \text{{result of the integral}} \), we multiply each result as follows:1. For \(a = 10\), final result is \(\frac{1.4711}{\pi} \approx 0.4680\)2. For \(a = 50\), final result is \(\frac{1.5508}{\pi} \approx 0.4936\)3. For \(a = 100\), final result is \(\frac{1.5608}{\pi} \approx 0.4971\)

Key Concepts

Arctangent Function

The arctangent function, often written as \( \tan^{-1}(x) \) or \( \arctan(x) \), is the inverse of the tangent function. It plays a crucial role in trigonometry and calculus, particularly when integrating functions like \((1+x^2)^{-1}\). This function has a range from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), covering all the angles that can produce real number inputs when operating with the tangent function.

• Inverse Nature: Since it reverses the operation of the tangent, you can use \( \tan(\tan^{-1}(x)) = x \) solving for angles.
• Derivative: The derivative of the arctangent function is \( (1+x^2)^{-1} \), which is why the indefinite integral of \((1+x^2)^{-1}\) gives us the arctangent.
• Practical Applications: It is frequently used in integration processes, especially when encountering integrals that let you simplify expressions by recognizing inverse trigonometric function derivatives.

Definite Integral

A definite integral is a type of integral that computes the accumulation of quantities, such as areas under a curve, over a specific interval \([a, b]\). With definite integrals, the result is a number rather than another function.

Some key details include:

• Limit of Integration: Defined between two limits, \(a\) and \(b\), these represent the start and end points of the interval.
• Fundamental Theorem of Calculus: This theorem connects differentiation and integration, explaining how the definite integral of a function over an interval can be calculated using its antiderivative.
• Application to Problems: In the given problem, the definite integral \(\int_{0}^{a} (1+x^2)^{-1} \, dx\) is evaluated by substituting limits into the antiderivative \(\tan^{-1}(x)\), giving \(\tan^{-1}(a) - \tan^{-1}(0)\).

The definite integral can help us understand various real-world phenomena, like calculating distances, areas, or even probabilities.

Indefinite Integral

The indefinite integral, also known as an antiderivative, is the opposite operation of differentiation. It represents a family of functions and is symbolized without specific boundaries. In general, if \(F(x)\) is an antiderivative of \(f(x)\), we write:\[\int f(x) \, dx = F(x) + C\]where \(C\) is the constant of integration.

The considerations involved are:

• Purpose: Indefinite integrals are useful for finding the most general form of the antiderivative before any specific values are evaluated.
• Integration Constants: The constant \(C\) accounts for all possible antiderivatives, as small variations can be adjusted by constants during derivation.
• Example Application: From the given problem, the indefinite integral of \((1+x^2)^{-1}\) is \(\tan^{-1}(x) + C\).

Understanding indefinite integrals is key to grasping the transition from general function forms to specific evaluations under given conditions.

Calculus

Calculus is a major branch of mathematics focusing on change and motion, with a particular emphasis on derivatives and integrals. It provides a framework for modeling systems and solving problems in sciences and engineering.

Key elements include:

• Derivatives: They describe how a function changes at a point and are foundational for understanding motion, growth, and optimization problems.
• Integrals: As the opposite of derivatives, integrals allow us to calculate areas, volumes, and other properties related to accumulation.
• Limits and Continuity: These concepts underpin the whole topic, with limits defining the instantaneous rate of change understood in derivatives, and continuity ensuring smooth function behavior.
• Real-world Expressions: Calculus models everything from planetary orbits and engineering stresses to economic risk assessments.

Through calculus, we can analyze and predict the behavior of dynamic systems, making it a pivotal tool in both theoretical and applied mathematics.