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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![**Topic: Calculating Surface Area of a Solid of Revolution**
When studying calculus, an interesting application involves computing the area of a surface that is generated by revolving a given curve around an axis. This concept is crucial in engineering, physics, and various fields of science.
---
**Problem Statement:**
Find the area of the surface generated when the given curve is revolved about the given axis.
Curve: \( y = x^3 \), for \( 0 \leq x \leq 1 \)
Axis of revolution: about the x-axis
**Solution Steps:**
1. Use the formula for the surface area \( S \) of a solid of revolution about the x-axis:
\[ S = 2\pi \int_{a}^{b} y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
2. Given \( y = x^3 \), first calculate the derivative \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = 3x^2 \]
3. Substitute \( y \) and \( \frac{dy}{dx} \) into the surface area formula:
\[ S = 2\pi \int_{0}^{1} x^3 \sqrt{1 + (3x^2)^2} \, dx \]
\[ S = 2\pi \int_{0}^{1} x^3 \sqrt{1 + 9x^4} \, dx \]
4. Solve the integral (this will typically require substitution or numerical methods).
**Your task:**
Calculate the exact surface area, ensuring to type your answer using \( \pi \) as needed.
**Answer Submission:**
The surface area is \( \boxed{\ \ \ \ \ \ \ } \) square units.
(Type an exact answer, using \( \pi \) as needed.)
---
After solving, you can check your answer by entering it in the provided answer box and clicking "Check Answer."
**Example for Practice:**
Try finding the surface area of other curves revolving about different axes to strengthen your understanding.
**Note:**
This problem incorporates key calculus concepts such as derivatives, integrals, and the application of these in geometrical contexts. Ensure thorough understanding and practice similar problems for mastery.
---
**End of Lesson**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F848d2852-24a2-4a72-a87d-6935de004e55%2F9c0dc47a-7989-46ed-b240-4ff0795eeaa5%2F2bd1ww.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Calculating Surface Area of a Solid of Revolution**
When studying calculus, an interesting application involves computing the area of a surface that is generated by revolving a given curve around an axis. This concept is crucial in engineering, physics, and various fields of science.
---
**Problem Statement:**
Find the area of the surface generated when the given curve is revolved about the given axis.
Curve: \( y = x^3 \), for \( 0 \leq x \leq 1 \)
Axis of revolution: about the x-axis
**Solution Steps:**
1. Use the formula for the surface area \( S \) of a solid of revolution about the x-axis:
\[ S = 2\pi \int_{a}^{b} y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
2. Given \( y = x^3 \), first calculate the derivative \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = 3x^2 \]
3. Substitute \( y \) and \( \frac{dy}{dx} \) into the surface area formula:
\[ S = 2\pi \int_{0}^{1} x^3 \sqrt{1 + (3x^2)^2} \, dx \]
\[ S = 2\pi \int_{0}^{1} x^3 \sqrt{1 + 9x^4} \, dx \]
4. Solve the integral (this will typically require substitution or numerical methods).
**Your task:**
Calculate the exact surface area, ensuring to type your answer using \( \pi \) as needed.
**Answer Submission:**
The surface area is \( \boxed{\ \ \ \ \ \ \ } \) square units.
(Type an exact answer, using \( \pi \) as needed.)
---
After solving, you can check your answer by entering it in the provided answer box and clicking "Check Answer."
**Example for Practice:**
Try finding the surface area of other curves revolving about different axes to strengthen your understanding.
**Note:**
This problem incorporates key calculus concepts such as derivatives, integrals, and the application of these in geometrical contexts. Ensure thorough understanding and practice similar problems for mastery.
---
**End of Lesson**
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