Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the graphs of the given equations about the y-axis. y = 5√√x, y = 0, x = 1
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the graphs of the given equations about the y-axis. y = 5√√x, y = 0, x = 1
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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![**Finding the Volume Using the Method of Cylindrical Shells**
To determine the volume generated by rotating the region bounded by the curves about the y-axis, apply the method of cylindrical shells. The equations provided for the curves are:
\[ y = 5\sqrt[3]{x}, \quad y = 0, \quad x = 1 \]
### Step-by-Step Solution:
1. **Identify the Boundaries:**
- The first equation \( y = 5\sqrt[3]{x} \) represents a curve.
- The second equation \( y = 0 \) represents the x-axis.
- The third boundary \( x = 1 \) is a vertical line.
2. **Setup for the Method of Cylindrical Shells:**
- When rotating around the y-axis, the height of each cylindrical shell is given by the function \( y = 5\sqrt[3]{x} \).
- The radius of each cylindrical shell is \( x \).
3. **Volume of a Single Shell:**
- The volume of a shell is given by \( 2\pi \times (\text{radius}) \times (\text{height}) \times (\text{thickness}) \).
- The thickness here is \( dx \).
- Thus, the volume element is \( dV = 2\pi x \cdot 5\sqrt[3]{x} \cdot dx \).
4. **Integrate to Find the Total Volume:**
- Integrate with respect to \( x \) from 0 to 1.
\[
V = \int_{0}^{1} 2\pi x \cdot 5\sqrt[3]{x} \, dx
\]
5. **Simplify the Integral:**
\[
V = 10\pi \int_{0}^{1} x \cdot x^{1/3} \, dx = 10\pi \int_{0}^{1} x^{4/3} \, dx
\]
6. **Evaluate the Integral:**
- The integral \( \int x^{4/3} \, dx \) can be evaluated using the power rule:
\[
\int x^{4/3} \, dx = \frac{3}{7} x^{7/3}
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Transcribed Image Text:**Finding the Volume Using the Method of Cylindrical Shells**
To determine the volume generated by rotating the region bounded by the curves about the y-axis, apply the method of cylindrical shells. The equations provided for the curves are:
\[ y = 5\sqrt[3]{x}, \quad y = 0, \quad x = 1 \]
### Step-by-Step Solution:
1. **Identify the Boundaries:**
- The first equation \( y = 5\sqrt[3]{x} \) represents a curve.
- The second equation \( y = 0 \) represents the x-axis.
- The third boundary \( x = 1 \) is a vertical line.
2. **Setup for the Method of Cylindrical Shells:**
- When rotating around the y-axis, the height of each cylindrical shell is given by the function \( y = 5\sqrt[3]{x} \).
- The radius of each cylindrical shell is \( x \).
3. **Volume of a Single Shell:**
- The volume of a shell is given by \( 2\pi \times (\text{radius}) \times (\text{height}) \times (\text{thickness}) \).
- The thickness here is \( dx \).
- Thus, the volume element is \( dV = 2\pi x \cdot 5\sqrt[3]{x} \cdot dx \).
4. **Integrate to Find the Total Volume:**
- Integrate with respect to \( x \) from 0 to 1.
\[
V = \int_{0}^{1} 2\pi x \cdot 5\sqrt[3]{x} \, dx
\]
5. **Simplify the Integral:**
\[
V = 10\pi \int_{0}^{1} x \cdot x^{1/3} \, dx = 10\pi \int_{0}^{1} x^{4/3} \, dx
\]
6. **Evaluate the Integral:**
- The integral \( \int x^{4/3} \, dx \) can be evaluated using the power rule:
\[
\int x^{4/3} \, dx = \frac{3}{7} x^{7/3}
\]
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