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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![**Example Problem: Evaluating an Integral using the Fundamental Theorem of Calculus**
Given the following conditions:
- \( F(4) = 4 \)
- \( F(10) = -9 \)
- \( F'(x) = f(x) \)
Determine the value of the integral:
\[ \int_{8}^{8} f(x) \, dx \]
**Explanation and Solution:**
From the given information, \( F'(x) = f(x) \), we can use the Fundamental Theorem of Calculus to evaluate the integral. The theorem states:
\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]
However, in this problem, we are asked to evaluate the integral from 8 to 8. For any continuous function \( f(x) \):
\[ \int_{a}^{a} f(x) \, dx \]
The above integral evaluates to 0 because there is no interval over which to integrate.
Therefore,
\[ \boxed{0} \]
is the value of the integral \( \int_{8}^{8} f(x) \, dx \).
This principle is important in calculus, demonstrating that integrating a function over an interval where the upper and lower limits are the same always results in zero, regardless of the behavior of the function within that interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffdc195bb-9009-47db-866e-65b6514806e8%2Fbad08bb8-0d5b-4169-bea3-ce9a1283ac7b%2Far6uhx6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Example Problem: Evaluating an Integral using the Fundamental Theorem of Calculus**
Given the following conditions:
- \( F(4) = 4 \)
- \( F(10) = -9 \)
- \( F'(x) = f(x) \)
Determine the value of the integral:
\[ \int_{8}^{8} f(x) \, dx \]
**Explanation and Solution:**
From the given information, \( F'(x) = f(x) \), we can use the Fundamental Theorem of Calculus to evaluate the integral. The theorem states:
\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]
However, in this problem, we are asked to evaluate the integral from 8 to 8. For any continuous function \( f(x) \):
\[ \int_{a}^{a} f(x) \, dx \]
The above integral evaluates to 0 because there is no interval over which to integrate.
Therefore,
\[ \boxed{0} \]
is the value of the integral \( \int_{8}^{8} f(x) \, dx \).
This principle is important in calculus, demonstrating that integrating a function over an interval where the upper and lower limits are the same always results in zero, regardless of the behavior of the function within that interval.
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