Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Riemann Sum
Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.
Riemann Integral
Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.
Question
![### Example Problem: Contour Integration for Real Definite Integrals
**Problem 5:**
Use contour integration to evaluate the real definite integral:
\[ \int_{0}^{2\pi} \frac{d\theta}{17 - 8\cos(\theta)}. \]
**Explanation:**
In this problem, you are tasked with evaluating a definite integral using contour integration, a method from complex analysis. The expression you need to integrate is dependent on trigonometric functions and defined in the interval from \(0\) to \(2\pi\).
### Step-by-Step Solution
To solve this problem using contour integration, follow these steps:
1. **Express the integral in the complex plane:**
- Replace \(\cos(\theta)\) with \( \frac{e^{i\theta} + e^{-i\theta}}{2} \).
2. **Identify the complex function:**
- Simplify the obtained expression to identify the complex function inside the integral.
3. **Apply the contour integration:**
- Use the residue theorem or other complex analysis tools to evaluate the integral.
4. **Translate the result back to the real integral:**
- Ensure the solution corresponds to the initial real integral.
For educational purposes, this example can guide students through the complexities of integrating with complex functions and highlight the practical application of intricate contour integration techniques.
Further explanations or the full derivation can be explored in-depth in advanced calculus or complex analysis textbooks.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8d14a18-18b6-4b9f-91b0-8c8acc9b34e2%2Fbf3bf785-bdcb-4260-8d4f-3a87c70a3570%2Fm5j5cba_processed.png&w=3840&q=75)
Transcribed Image Text:### Example Problem: Contour Integration for Real Definite Integrals
**Problem 5:**
Use contour integration to evaluate the real definite integral:
\[ \int_{0}^{2\pi} \frac{d\theta}{17 - 8\cos(\theta)}. \]
**Explanation:**
In this problem, you are tasked with evaluating a definite integral using contour integration, a method from complex analysis. The expression you need to integrate is dependent on trigonometric functions and defined in the interval from \(0\) to \(2\pi\).
### Step-by-Step Solution
To solve this problem using contour integration, follow these steps:
1. **Express the integral in the complex plane:**
- Replace \(\cos(\theta)\) with \( \frac{e^{i\theta} + e^{-i\theta}}{2} \).
2. **Identify the complex function:**
- Simplify the obtained expression to identify the complex function inside the integral.
3. **Apply the contour integration:**
- Use the residue theorem or other complex analysis tools to evaluate the integral.
4. **Translate the result back to the real integral:**
- Ensure the solution corresponds to the initial real integral.
For educational purposes, this example can guide students through the complexities of integrating with complex functions and highlight the practical application of intricate contour integration techniques.
Further explanations or the full derivation can be explored in-depth in advanced calculus or complex analysis textbooks.
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