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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![**Educational Content: Calculating Area Using Partial Fractions**
**Problem Statement:**
Use the method of partial fractions to find the area bounded by the curve given by the equation:
\[ y = \frac{2x - 8}{x^2 - 5x + 6} \]
The area is bounded by the x-axis, \( x = 9 \), and \( x = 13 \). Show all work.
**Solution Explanation:**
To solve this problem, we follow these steps:
1. **Identify the Fraction to Decompose:**
The function is in the form of a rational function \( \frac{2x - 8}{x^2 - 5x + 6} \).
2. **Factor the Denominator:**
The quadratic denominator \( x^2 - 5x + 6 \) can be factored into \((x - 2)(x - 3)\).
3. **Setup Partial Fractions:**
The expression can be decomposed as:
\[
\frac{2x - 8}{(x - 2)(x - 3)} = \frac{A}{x - 2} + \frac{B}{x - 3}
\]
Here, \( A \) and \( B \) are constants we need to determine.
4. **Solve for Constants \( A \) and \( B \):**
Multiply through by the common denominator to clear fractions:
\[
2x - 8 = A(x - 3) + B(x - 2)
\]
Expand and equate coefficients to solve for \( A \) and \( B \).
5. **Integrate to Find the Area:**
Once the partial fractions are found, integrate each fraction separately between the limits \( x = 9 \) and \( x = 13 \) to find the bounded area.
6. **Apply Calculus to Find Definite Integrals:**
Use techniques of integration to evaluate the area under the curve within the specified limits.
**Conclusion:**
By using partial fractions, we break down complex rational expressions into simpler forms, making it easier to integrate. This method is particularly useful in finding areas under curves, providing a systematic approach to solving complex calculus problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F281e2029-de70-4b7c-8ebc-bf0179904e66%2Fbd60f706-d4a4-4269-91be-d5cb33f015b9%2F76xyvi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Content: Calculating Area Using Partial Fractions**
**Problem Statement:**
Use the method of partial fractions to find the area bounded by the curve given by the equation:
\[ y = \frac{2x - 8}{x^2 - 5x + 6} \]
The area is bounded by the x-axis, \( x = 9 \), and \( x = 13 \). Show all work.
**Solution Explanation:**
To solve this problem, we follow these steps:
1. **Identify the Fraction to Decompose:**
The function is in the form of a rational function \( \frac{2x - 8}{x^2 - 5x + 6} \).
2. **Factor the Denominator:**
The quadratic denominator \( x^2 - 5x + 6 \) can be factored into \((x - 2)(x - 3)\).
3. **Setup Partial Fractions:**
The expression can be decomposed as:
\[
\frac{2x - 8}{(x - 2)(x - 3)} = \frac{A}{x - 2} + \frac{B}{x - 3}
\]
Here, \( A \) and \( B \) are constants we need to determine.
4. **Solve for Constants \( A \) and \( B \):**
Multiply through by the common denominator to clear fractions:
\[
2x - 8 = A(x - 3) + B(x - 2)
\]
Expand and equate coefficients to solve for \( A \) and \( B \).
5. **Integrate to Find the Area:**
Once the partial fractions are found, integrate each fraction separately between the limits \( x = 9 \) and \( x = 13 \) to find the bounded area.
6. **Apply Calculus to Find Definite Integrals:**
Use techniques of integration to evaluate the area under the curve within the specified limits.
**Conclusion:**
By using partial fractions, we break down complex rational expressions into simpler forms, making it easier to integrate. This method is particularly useful in finding areas under curves, providing a systematic approach to solving complex calculus problems.
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