Use the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis. y = 1 x21 y = 0, x = 4, x = 9
Use the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis. y = 1 x21 y = 0, x = 4, x = 9
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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![### Problem 2
**Objective:**
Use the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis.
**Given:**
\[ y = \frac{1}{x^2} \]
\[ y = 0 \]
\[ x = 4 \]
\[ x = 9 \]
**Approach:**
To find the volume of the solid using the shell method, we need to set up and evaluate the integral. The shell method formula for a solid of revolution around the y-axis is:
\[
V = 2\pi \int_{a}^{b} \left(\text{radius}\right) \left(\text{height}\right) \, dx
\]
In this problem:
- The "radius" is the distance from the y-axis, which is \(x\).
- The "height" is the value of the function \(y = \frac{1}{x^2}\).
Substitute the limits of integration \(a = 4\) and \(b = 9\) into the formula.
Therefore, the volume \(V\) is given by:
\[
V = 2\pi \int_{4}^{9} x \left( \frac{1}{x^2} \right) \, dx
\]
Simplify the integral:
\[
V = 2\pi \int_{4}^{9} \frac{1}{x} \, dx
\]
Evaluate the integral:
\[
V = 2\pi \left[ \ln|x| \right]_{4}^{9}
\]
Calculating the definite integral:
\[
V = 2\pi \left( \ln 9 - \ln 4 \right)
\]
Simplify using properties of logarithms:
\[
V = 2\pi \ln\left(\frac{9}{4}\right)
\]
Thus, the volume of the solid is:
\[
V = 2\pi \ln\left(\frac{9}{4}\right)
\]
**Conclusion:**
The volume of the solid generated by revolving the given region about the y-axis is \( 2\pi \ln\left(\frac{9}{4}\right) \) cubic units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff23b3b8c-26d4-489d-ae68-c71751627f4f%2F6987ba19-c1b1-469b-97fe-cce632ff1ac2%2F56plue9_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 2
**Objective:**
Use the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis.
**Given:**
\[ y = \frac{1}{x^2} \]
\[ y = 0 \]
\[ x = 4 \]
\[ x = 9 \]
**Approach:**
To find the volume of the solid using the shell method, we need to set up and evaluate the integral. The shell method formula for a solid of revolution around the y-axis is:
\[
V = 2\pi \int_{a}^{b} \left(\text{radius}\right) \left(\text{height}\right) \, dx
\]
In this problem:
- The "radius" is the distance from the y-axis, which is \(x\).
- The "height" is the value of the function \(y = \frac{1}{x^2}\).
Substitute the limits of integration \(a = 4\) and \(b = 9\) into the formula.
Therefore, the volume \(V\) is given by:
\[
V = 2\pi \int_{4}^{9} x \left( \frac{1}{x^2} \right) \, dx
\]
Simplify the integral:
\[
V = 2\pi \int_{4}^{9} \frac{1}{x} \, dx
\]
Evaluate the integral:
\[
V = 2\pi \left[ \ln|x| \right]_{4}^{9}
\]
Calculating the definite integral:
\[
V = 2\pi \left( \ln 9 - \ln 4 \right)
\]
Simplify using properties of logarithms:
\[
V = 2\pi \ln\left(\frac{9}{4}\right)
\]
Thus, the volume of the solid is:
\[
V = 2\pi \ln\left(\frac{9}{4}\right)
\]
**Conclusion:**
The volume of the solid generated by revolving the given region about the y-axis is \( 2\pi \ln\left(\frac{9}{4}\right) \) cubic units.
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