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Magazine Chapter 8: Problem 42
Approximate \(\int_{0}^{a} \frac{1}{\pi}\left(1+x^{2}\right)^{-1} d x\) for \(a=10,50\), and \(100 .\)
Short Answer
Expert verified
Approximate integrals: for 10, 0.4680; for 50, 0.4936; for 100, 0.4971.
Step by step solution
01
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.
02
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 \(\tan^{-1}(x)\).
03
Compute the Indefinite Integral
Calculate \(\int (1+x^2)^{-1} \, dx = \tan^{-1}(x) + C\), where \(C\) is a constant of integration.
04
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 \(\tan^{-1}(0) = 0\).
05
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.
06
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 \(\frac{1.5608}{\pi} \approx 0.4971\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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 \(\tan^{-1}(a) - \tan^{-1}(0)\).
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:
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 \(\tan^{-1}(x) + C\).
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:
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.
