O Choose a coordinate system and set up a triple integral to compute the volume of the solid.
O Choose a coordinate system and set up a triple integral to compute the volume of the solid.
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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![**Title: Setting Up a Triple Integral to Compute the Volume of a Solid**
**Objective:**
Learn how to choose a coordinate system and set up a triple integral to determine the volume of a given solid.
**Text:**
Choose a coordinate system and set up a triple integral to compute the volume of the solid.
**Diagram Explanation:**
The diagram depicts a three-dimensional solid resembling a quarter-cylinder. The structure can be described in a Cartesian coordinate system with:
- The x, y, and z axes are indicated.
- The solid extends vertically from z=0 to z=4.
- Its base is a quarter-circle with a radius of 2 units, lying in the xy-plane.
- The height, from the base to the top, is 3 units.
**Steps to Set Up the Triple Integral:**
1. **Coordinate System:** Use cylindrical coordinates, advantageous for solids with circular boundaries.
2. **Integration Limits:**
- For \( r \) (radius): \( 0 \leq r \leq 2 \)
- For \( \theta \) (angle): \( 0 \leq \theta \leq \frac{\pi}{2} \)
- For \( z \): \( 0 \leq z \leq 3 \)
3. **Triple Integral Setup:**
\[
V = \int_{0}^{\frac{\pi}{2}} \int_{0}^{2} \int_{0}^{3} r \, dz \, dr \, d\theta
\]
**Conclusion:**
By setting up this triple integral, you can compute the volume of the solid efficiently using cylindrical coordinates. This method simplifies the integration process when dealing with rotationally symmetric solids.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faaadfa70-4428-4dd8-8c22-944f62d3e464%2F04efab22-58f7-4317-854c-2f27744c298e%2Fl0jdy0z_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Setting Up a Triple Integral to Compute the Volume of a Solid**
**Objective:**
Learn how to choose a coordinate system and set up a triple integral to determine the volume of a given solid.
**Text:**
Choose a coordinate system and set up a triple integral to compute the volume of the solid.
**Diagram Explanation:**
The diagram depicts a three-dimensional solid resembling a quarter-cylinder. The structure can be described in a Cartesian coordinate system with:
- The x, y, and z axes are indicated.
- The solid extends vertically from z=0 to z=4.
- Its base is a quarter-circle with a radius of 2 units, lying in the xy-plane.
- The height, from the base to the top, is 3 units.
**Steps to Set Up the Triple Integral:**
1. **Coordinate System:** Use cylindrical coordinates, advantageous for solids with circular boundaries.
2. **Integration Limits:**
- For \( r \) (radius): \( 0 \leq r \leq 2 \)
- For \( \theta \) (angle): \( 0 \leq \theta \leq \frac{\pi}{2} \)
- For \( z \): \( 0 \leq z \leq 3 \)
3. **Triple Integral Setup:**
\[
V = \int_{0}^{\frac{\pi}{2}} \int_{0}^{2} \int_{0}^{3} r \, dz \, dr \, d\theta
\]
**Conclusion:**
By setting up this triple integral, you can compute the volume of the solid efficiently using cylindrical coordinates. This method simplifies the integration process when dealing with rotationally symmetric solids.
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