Find the volume of the solid obtained by rotating the region bounded by y = 4x², x = 1, x = 2 and y = 0, about the x-axis. (1,2,0) V = Xo311.64599123611
Find the volume of the solid obtained by rotating the region bounded by y = 4x², x = 1, x = 2 and y = 0, about the x-axis. (1,2,0) V = Xo311.64599123611
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Calculating the Volume of a Solid of Revolution
**Problem Statement:**
Find the volume of the solid obtained by rotating the region bounded by the equations \( y = 4x^2 \), \( x = 1 \), \( x = 2 \), and \( y = 0 \) about the \( x \)-axis.
**Solution:**
The volume \( V \) of the solid formed by rotating the given region about the \( x \)-axis can be calculated using the disk method. The general formula for the volume using the disk method is:
\[ V = \pi \int_{a}^{b} [R(x)]^2 \, dx \]
Where \( R(x) \) is the distance from the \( x \)-axis to the curve. In this case, \( R(x) \) is given by the function \( y = 4x^2 \) and the bounds of integration are from \( x = 1 \) to \( x = 2 \).
Therefore, the volume \( V \) can be expressed as:
\[ V = \pi \int_{1}^{2} (4x^2)^2 \, dx \]
Simplifying the integrand:
\[ V = \pi \int_{1}^{2} 16x^4 \, dx \]
Now, integrate \( 16x^4 \):
\[ V = 16\pi \int_{1}^{2} x^4 \, dx \]
\[ V = 16\pi \left[\frac{x^5}{5}\right]_{1}^{2} \]
Evaluate the definite integral:
\[ V = 16\pi \left[\frac{2^5}{5} - \frac{1^5}{5}\right] \]
\[ V = 16\pi \left[\frac{32}{5} - \frac{1}{5}\right] \]
\[ V = 16\pi \left[\frac{31}{5}\right] \]
\[ V = \frac{496\pi}{5} \]
Using the approximate value of \(\pi \approx 3.14159265359\):
\[ V \approx \frac{496 \cdot 3.14159265359}{5} \]
\[ V \approx 311.64599123611 \]
**Final Answer:**
The volume of the solid obtained by rotating the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d56d87b-1ee4-40b0-9831-f06655864a4b%2F2c03d248-8883-4d8a-97ca-7c3b82702619%2Fr7hmua_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating the Volume of a Solid of Revolution
**Problem Statement:**
Find the volume of the solid obtained by rotating the region bounded by the equations \( y = 4x^2 \), \( x = 1 \), \( x = 2 \), and \( y = 0 \) about the \( x \)-axis.
**Solution:**
The volume \( V \) of the solid formed by rotating the given region about the \( x \)-axis can be calculated using the disk method. The general formula for the volume using the disk method is:
\[ V = \pi \int_{a}^{b} [R(x)]^2 \, dx \]
Where \( R(x) \) is the distance from the \( x \)-axis to the curve. In this case, \( R(x) \) is given by the function \( y = 4x^2 \) and the bounds of integration are from \( x = 1 \) to \( x = 2 \).
Therefore, the volume \( V \) can be expressed as:
\[ V = \pi \int_{1}^{2} (4x^2)^2 \, dx \]
Simplifying the integrand:
\[ V = \pi \int_{1}^{2} 16x^4 \, dx \]
Now, integrate \( 16x^4 \):
\[ V = 16\pi \int_{1}^{2} x^4 \, dx \]
\[ V = 16\pi \left[\frac{x^5}{5}\right]_{1}^{2} \]
Evaluate the definite integral:
\[ V = 16\pi \left[\frac{2^5}{5} - \frac{1^5}{5}\right] \]
\[ V = 16\pi \left[\frac{32}{5} - \frac{1}{5}\right] \]
\[ V = 16\pi \left[\frac{31}{5}\right] \]
\[ V = \frac{496\pi}{5} \]
Using the approximate value of \(\pi \approx 3.14159265359\):
\[ V \approx \frac{496 \cdot 3.14159265359}{5} \]
\[ V \approx 311.64599123611 \]
**Final Answer:**
The volume of the solid obtained by rotating the
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