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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![### Calculus Problem: Finding the Value of a Definite Integral
**Problem 3:**
Evaluate the definite integral:
\[ \int_{5}^{11} \frac{1}{x^2} \, dx \]
### Explanation:
This problem asks for the value of the definite integral of the function \(\frac{1}{x^2}\) from \(x = 5\) to \(x = 11\). To solve this, you need to apply the rules of integration and evaluate the integral within the given limits.
First, recognize that \(\frac{1}{x^2}\) can be written as \(x^{-2}\). The indefinite integral of \(x^{-2}\) is found using the power rule for integration, which states:
\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{(where \(n \neq -1\))}
\]
For \(n = -2\):
\[
\int x^{-2} \, dx = \int x^{-2} \, dx = \frac{x^{-2+1}}{-2+1} + C = \frac{x^{-1}}{-1} + C = -\frac{1}{x} + C
\]
Next, apply the limits of integration from 5 to 11:
\[
\int_{5}^{11} x^{-2} \, dx = \left[ -\frac{1}{x} \right]_{5}^{11} = \left( -\frac{1}{11} \right) - \left( -\frac{1}{5} \right) = -\frac{1}{11} + \frac{1}{5}
\]
To combine the fractions, find a common denominator:
\[
-\frac{1}{11} + \frac{1}{5} = -\frac{5}{55} + \frac{11}{55} = \frac{11 - 5}{55} = \frac{6}{55}
\]
So the value of the definite integral is:
\[
\frac{6}{55}
\]
### Result:
The value of the integral \( \int_{5}^{11} \frac{1}{x^2} \, dx \) is \(\frac{6}{55}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F469aef3b-31ef-41c9-aa0d-266ee90c84d9%2F563a0eee-5180-4c35-8126-a00413e5eb48%2F4nqm0x_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculus Problem: Finding the Value of a Definite Integral
**Problem 3:**
Evaluate the definite integral:
\[ \int_{5}^{11} \frac{1}{x^2} \, dx \]
### Explanation:
This problem asks for the value of the definite integral of the function \(\frac{1}{x^2}\) from \(x = 5\) to \(x = 11\). To solve this, you need to apply the rules of integration and evaluate the integral within the given limits.
First, recognize that \(\frac{1}{x^2}\) can be written as \(x^{-2}\). The indefinite integral of \(x^{-2}\) is found using the power rule for integration, which states:
\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{(where \(n \neq -1\))}
\]
For \(n = -2\):
\[
\int x^{-2} \, dx = \int x^{-2} \, dx = \frac{x^{-2+1}}{-2+1} + C = \frac{x^{-1}}{-1} + C = -\frac{1}{x} + C
\]
Next, apply the limits of integration from 5 to 11:
\[
\int_{5}^{11} x^{-2} \, dx = \left[ -\frac{1}{x} \right]_{5}^{11} = \left( -\frac{1}{11} \right) - \left( -\frac{1}{5} \right) = -\frac{1}{11} + \frac{1}{5}
\]
To combine the fractions, find a common denominator:
\[
-\frac{1}{11} + \frac{1}{5} = -\frac{5}{55} + \frac{11}{55} = \frac{11 - 5}{55} = \frac{6}{55}
\]
So the value of the definite integral is:
\[
\frac{6}{55}
\]
### Result:
The value of the integral \( \int_{5}^{11} \frac{1}{x^2} \, dx \) is \(\frac{6}{55}\).
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