Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve about the given axis. y=e*, 1 ≤ y ≤ 7; y-axis Select the correct answer. In (7) 2x √27√1 + e² dx 0 In (7) 2x √ 27x√1 + e -e ²x dx 0 7 O{2+²5√/1 + a²dx e 2x O]2+√/1 + ²x e d
Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve about the given axis. y=e*, 1 ≤ y ≤ 7; y-axis Select the correct answer. In (7) 2x √27√1 + e² dx 0 In (7) 2x √ 27x√1 + e -e ²x dx 0 7 O{2+²5√/1 + a²dx e 2x O]2+√/1 + ²x e d
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 Surface Area of a Rotated Curve
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To compute the integral for the surface area of the curve rotated about the given axis, follow the instructions below.
**Curve Equation:**
\[ y = e^x \quad,\quad 1 \leq y \leq 7 \]
**Axis of Rotation:**
\[ y\text{-axis} \]
You need to set up, but do not evaluate, the integral to find the area of the surface created by rotating the curve about the y-axis according to the given parameters.
### Choices for Surface Area Integral:
Select the correct integral expression from the choices below:
1. \[ \int_{0}^{\ln(7)} 2\pi\sqrt{1 + e^{2x}} \, dx \]
2. \[ \int_{0}^{\ln(7)} 2\pi x \sqrt{1 + e^{2x}} \, dx \]
3. \[ \int_{1}^{7} 2\pi e^x \sqrt{1 + e^{2x}} \, dx \]
4. \[ \int_{1}^{7} 2\pi x \sqrt{1 + e^{2x}} \, dx \]
Make sure to select the correct integral setup that matches the problem's criteria.
### Explanation of the Integral Setup:
- The surface area \(S\) of a curve \(y=f(x)\) rotated about the y-axis from \(y=a\) to \(y=b\) is given by:
\[ S = \int_{a}^{b} 2\pi x \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
In the context of this problem, you will need to determine which limits of integration correctly map to the given \(y\)-values and the correct form of the radius and integrand components.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24e8bfde-46ac-4815-81bc-d8be1d9c4f35%2F69cf2a96-d2d2-460c-af5a-61f9897ac703%2Fz78m4ip_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating the Surface Area of a Rotated Curve
---
To compute the integral for the surface area of the curve rotated about the given axis, follow the instructions below.
**Curve Equation:**
\[ y = e^x \quad,\quad 1 \leq y \leq 7 \]
**Axis of Rotation:**
\[ y\text{-axis} \]
You need to set up, but do not evaluate, the integral to find the area of the surface created by rotating the curve about the y-axis according to the given parameters.
### Choices for Surface Area Integral:
Select the correct integral expression from the choices below:
1. \[ \int_{0}^{\ln(7)} 2\pi\sqrt{1 + e^{2x}} \, dx \]
2. \[ \int_{0}^{\ln(7)} 2\pi x \sqrt{1 + e^{2x}} \, dx \]
3. \[ \int_{1}^{7} 2\pi e^x \sqrt{1 + e^{2x}} \, dx \]
4. \[ \int_{1}^{7} 2\pi x \sqrt{1 + e^{2x}} \, dx \]
Make sure to select the correct integral setup that matches the problem's criteria.
### Explanation of the Integral Setup:
- The surface area \(S\) of a curve \(y=f(x)\) rotated about the y-axis from \(y=a\) to \(y=b\) is given by:
\[ S = \int_{a}^{b} 2\pi x \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
In the context of this problem, you will need to determine which limits of integration correctly map to the given \(y\)-values and the correct form of the radius and integrand components.
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