Set up, but do not evaluate, and integral to find the volume of the solid formed by rotating the region bounded by the graph of y = v1 – x²2, y = 0, and x = 0 about the line x = 2.
Set up, but do not evaluate, and integral to find the volume of the solid formed by rotating the region bounded by the graph of y = v1 – x²2, y = 0, and x = 0 about the line x = 2.
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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I would really appreciate any help with this one! This is calculus 2. Just set up an integral to find the volume of the solid given the credientials. I get very confused!
![**Problem Statement:**
Set up, but do not evaluate, an integral to find the volume of the solid formed by rotating the region bounded by the graph of \( y = \sqrt{1 - x^2} \), \( y = 0 \), and \( x = 0 \) about the line \( x = 2 \).
**Explanation:**
- The given function \( y = \sqrt{1 - x^2} \) represents the top half of a circle with radius 1 centered at the origin.
- The boundaries given include \( y = 0 \) (the x-axis) and \( x = 0 \) (the y-axis), defining a quarter circle in the first quadrant.
- The task is to find the volume of the solid generated by rotating this region about the vertical line \( x = 2 \), which lies to the right of the region.
**Steps for Setting Up the Integral:**
1. **Identify the Region:** The area of interest is a quarter circle in the first quadrant.
2. **Rotational Axis:** The entire region is rotated about the line \( x = 2 \).
3. **Method:** Use the method of cylindrical shells or washers/disks, depending on the preference for visualization:
- **Cylindrical Shells:** Consider an infinitesimal vertical strip at position \( x \) with height \( y = \sqrt{1 - x^2} \) and thickness \( dx \). The distance from the axis \( x = 2 \) is \( 2 - x \).
- **Washer/Disc Method:** The outer radius of rotation is \( 2 \), and the inner radius is \( 2 - x \).
4. **Setup of the Integral:** Express the volume \( V \) as an integral based on the chosen method.
**Cylindrical Shells Integral:**
\[
V = 2\pi \int_0^1 (2 - x) \sqrt{1 - x^2} \, dx
\]
**Washer Method:**
\[
V = \pi \int_0^{\pi/2} [(2^2) - (2 - \cos \theta)^2] \cdot \sin \theta \, d\theta
\]
*(Convert to a parameterized form if preferred using trigonometric substitution.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f7679a1-29bd-41dc-b06e-05edfd912e91%2F6fb479ca-df1b-4b21-a1cb-d56286a20e50%2Fb8n3eg8_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Set up, but do not evaluate, an integral to find the volume of the solid formed by rotating the region bounded by the graph of \( y = \sqrt{1 - x^2} \), \( y = 0 \), and \( x = 0 \) about the line \( x = 2 \).
**Explanation:**
- The given function \( y = \sqrt{1 - x^2} \) represents the top half of a circle with radius 1 centered at the origin.
- The boundaries given include \( y = 0 \) (the x-axis) and \( x = 0 \) (the y-axis), defining a quarter circle in the first quadrant.
- The task is to find the volume of the solid generated by rotating this region about the vertical line \( x = 2 \), which lies to the right of the region.
**Steps for Setting Up the Integral:**
1. **Identify the Region:** The area of interest is a quarter circle in the first quadrant.
2. **Rotational Axis:** The entire region is rotated about the line \( x = 2 \).
3. **Method:** Use the method of cylindrical shells or washers/disks, depending on the preference for visualization:
- **Cylindrical Shells:** Consider an infinitesimal vertical strip at position \( x \) with height \( y = \sqrt{1 - x^2} \) and thickness \( dx \). The distance from the axis \( x = 2 \) is \( 2 - x \).
- **Washer/Disc Method:** The outer radius of rotation is \( 2 \), and the inner radius is \( 2 - x \).
4. **Setup of the Integral:** Express the volume \( V \) as an integral based on the chosen method.
**Cylindrical Shells Integral:**
\[
V = 2\pi \int_0^1 (2 - x) \sqrt{1 - x^2} \, dx
\]
**Washer Method:**
\[
V = \pi \int_0^{\pi/2} [(2^2) - (2 - \cos \theta)^2] \cdot \sin \theta \, d\theta
\]
*(Convert to a parameterized form if preferred using trigonometric substitution.)
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