Compute the surface area of revolution about the x-axis over the interval [0, n] for y = 8 sin(x). (Use symbolic notation and fractions where needed.) S =
Compute the surface area of revolution about the x-axis over the interval [0, n] for y = 8 sin(x). (Use symbolic notation and fractions where needed.) S =
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...
Related questions
Question
![To compute the surface area of revolution about the x-axis for the function \( y = 8 \sin(x) \) over the interval \([0, \pi]\), we use the formula for the surface area \( S \) of a solid of revolution:
\[
S = \int_{a}^{b} 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
\]
For the function \( y = 8 \sin(x) \):
- The derivative \( \frac{dy}{dx} = 8 \cos(x) \).
Substitute \( y \) and \( \frac{dy}{dx} \) into the formula:
\[
S = \int_{0}^{\pi} 2\pi (8 \sin(x)) \sqrt{1 + (8 \cos(x))^2} \, dx
\]
This integral represents the surface area of the solid formed by revolving the curve \( y = 8 \sin(x) \) around the x-axis from \( x = 0 \) to \( x = \pi \).
The solution involves evaluating this integral, which may require numerical methods or further analytical techniques depending on the complexity of the expression inside the integral.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa297a8fb-6644-4f66-925a-05650f7247e5%2F1678394b-a768-4673-a767-89b1df2b9be0%2Fx7hopg6_processed.png&w=3840&q=75)
Transcribed Image Text:To compute the surface area of revolution about the x-axis for the function \( y = 8 \sin(x) \) over the interval \([0, \pi]\), we use the formula for the surface area \( S \) of a solid of revolution:
\[
S = \int_{a}^{b} 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
\]
For the function \( y = 8 \sin(x) \):
- The derivative \( \frac{dy}{dx} = 8 \cos(x) \).
Substitute \( y \) and \( \frac{dy}{dx} \) into the formula:
\[
S = \int_{0}^{\pi} 2\pi (8 \sin(x)) \sqrt{1 + (8 \cos(x))^2} \, dx
\]
This integral represents the surface area of the solid formed by revolving the curve \( y = 8 \sin(x) \) around the x-axis from \( x = 0 \) to \( x = \pi \).
The solution involves evaluating this integral, which may require numerical methods or further analytical techniques depending on the complexity of the expression inside the integral.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning