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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![### Calculating the Length of a Curve
To find the length \( L \) of the curve from \( x = 4 \) to \( x = 6 \), we use the formula for the arc length of a function:
\[
L = \int_{4}^{6} \sqrt{1 + \left( f'(x) \right)^2} \, dx
\]
In this example, this integral can be simplified to:
\[
L = \int_{4}^{6} \frac{1}{2} \left( x^2 + \frac{1}{x^2} \right) \, dx
\]
Next, we represent the result of integrating this expression as follows:
\[
L = \left( \boxed{\phantom{expression}} \right)_{4}^{6}
\]
Finally, solving the definite integral yields:
\[
L = \boxed{\phantom{express}}
\]
Note: The red cross indicates an incorrect or unresolved step in the solution process.
This set of equations demonstrates the method to find the length of the curve using definite integration and simplification of functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe1f37d79-53bb-43ed-93c0-29d608b5ceba%2F3c6d93e8-46d9-4edd-a1f3-d542206deae1%2Fkaxjlo_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating the Length of a Curve
To find the length \( L \) of the curve from \( x = 4 \) to \( x = 6 \), we use the formula for the arc length of a function:
\[
L = \int_{4}^{6} \sqrt{1 + \left( f'(x) \right)^2} \, dx
\]
In this example, this integral can be simplified to:
\[
L = \int_{4}^{6} \frac{1}{2} \left( x^2 + \frac{1}{x^2} \right) \, dx
\]
Next, we represent the result of integrating this expression as follows:
\[
L = \left( \boxed{\phantom{expression}} \right)_{4}^{6}
\]
Finally, solving the definite integral yields:
\[
L = \boxed{\phantom{express}}
\]
Note: The red cross indicates an incorrect or unresolved step in the solution process.
This set of equations demonstrates the method to find the length of the curve using definite integration and simplification of functions.
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