Find the area of the surface obtained by rotating the curve about the x-axis. X 름-빨 In x Select the correct answer. y = [1,720-81 In (7) - (In (7))² [359-361 n (7) - (In(7))²] [648-491n (7) - (In(7))²] [2,599-1001n (7) - (In(7))²] 1 ≤ x ≤7 7 2
Find the area of the surface obtained by rotating the curve about the x-axis. X 름-빨 In x Select the correct answer. y = [1,720-81 In (7) - (In (7))² [359-361 n (7) - (In(7))²] [648-491n (7) - (In(7))²] [2,599-1001n (7) - (In(7))²] 1 ≤ x ≤7 7 2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Surface Area of a Rotated Curve
To find the area of the surface obtained by rotating the curve about the x-axis, consider the curve defined by the equation:
\[ y = \frac{x^2}{4} - \frac{\ln x}{2}, \quad 1 \leq x \leq 7 \]
#### Question:
Select the correct answer for the area obtained by rotating the given curve about the x-axis.
The options are:
1. \(\frac{\pi}{4} \left[ 1,720 - 81 \ln (7) - (\ln (7))^2 \right]\)
2. \(\frac{\pi}{4} \left[ 359 - 36 \ln (7) - (\ln (7))^2 \right]\)
3. \(\frac{\pi}{4} \left[ 648 - 49 \ln (7) - (\ln (7))^2 \right]\)
4. \(\frac{\pi}{4} \left[ 2,599 - 100 \ln (7) - (\ln (7))^2 \right]\)
To solve this problem, we need to determine the correct expression for the area of the surface generated by rotating the given curve \( y = \frac{x^2}{4} - \frac{\ln x}{2} \) around the x-axis between \( x = 1 \) and \( x = 7 \).
### Explanation:
In solving problems of this type, we typically apply the formula for the surface area of a solid of revolution around the x-axis, given by:
\[ A = 2\pi \int_a^b y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
Where:
- \( y \) is the function describing the curve.
- \( \frac{dy}{dx} \) is the derivative of the function describing the curve.
- \( [a, b] \) is the interval over which we are integrating.
In this case:
- \( a = 1 \)
- \( b = 7 \)
- \( y = \frac{x^2}{4} - \frac{\ln x}{2} \)
You would compute \(\frac{dy}{dx}\), plug these into the surface area formula, and evaluate the integral to obtain the correct solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24e8bfde-46ac-4815-81bc-d8be1d9c4f35%2Feb730ccd-ebee-42ab-b5da-86de0cf3beaf%2Fsbq632n_processed.png&w=3840&q=75)
Transcribed Image Text:### Surface Area of a Rotated Curve
To find the area of the surface obtained by rotating the curve about the x-axis, consider the curve defined by the equation:
\[ y = \frac{x^2}{4} - \frac{\ln x}{2}, \quad 1 \leq x \leq 7 \]
#### Question:
Select the correct answer for the area obtained by rotating the given curve about the x-axis.
The options are:
1. \(\frac{\pi}{4} \left[ 1,720 - 81 \ln (7) - (\ln (7))^2 \right]\)
2. \(\frac{\pi}{4} \left[ 359 - 36 \ln (7) - (\ln (7))^2 \right]\)
3. \(\frac{\pi}{4} \left[ 648 - 49 \ln (7) - (\ln (7))^2 \right]\)
4. \(\frac{\pi}{4} \left[ 2,599 - 100 \ln (7) - (\ln (7))^2 \right]\)
To solve this problem, we need to determine the correct expression for the area of the surface generated by rotating the given curve \( y = \frac{x^2}{4} - \frac{\ln x}{2} \) around the x-axis between \( x = 1 \) and \( x = 7 \).
### Explanation:
In solving problems of this type, we typically apply the formula for the surface area of a solid of revolution around the x-axis, given by:
\[ A = 2\pi \int_a^b y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
Where:
- \( y \) is the function describing the curve.
- \( \frac{dy}{dx} \) is the derivative of the function describing the curve.
- \( [a, b] \) is the interval over which we are integrating.
In this case:
- \( a = 1 \)
- \( b = 7 \)
- \( y = \frac{x^2}{4} - \frac{\ln x}{2} \)
You would compute \(\frac{dy}{dx}\), plug these into the surface area formula, and evaluate the integral to obtain the correct solution.
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