18. Give a rough sketch of the solid that is enclosed by the paraboloids z = x² + y² and z = 18 - x² - y² and use your sketch to set up but DO NOT evaluate the triple that would represent the volume between the two curves.
18. Give a rough sketch of the solid that is enclosed by the paraboloids z = x² + y² and z = 18 - x² - y² and use your sketch to set up but DO NOT evaluate the triple that would represent the volume between the two curves.
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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![### Problem 18
**Task:**
Provide a rough sketch of the solid enclosed by the paraboloids \( z = x^2 + y^2 \) and \( z = 18 - x^2 - y^2 \). Use the sketch to set up, but **DO NOT** evaluate, the triple integral that represents the volume between the two surfaces.
**Explanation:**
1. **Paraboloids Description:**
- \( z = x^2 + y^2 \) describes an upward-opening paraboloid with its vertex at the origin \((0,0,0)\).
- \( z = 18 - x^2 - y^2 \) describes a downward-opening paraboloid with its vertex at \((0,0,18)\).
2. **Intersection:**
- To find where the two paraboloids intersect, set their equations equal: \( x^2 + y^2 = 18 - x^2 - y^2 \).
- Simplifying gives \( 2x^2 + 2y^2 = 18 \), or \( x^2 + y^2 = 9 \). This describes a circle of radius 3 in the \( xy \)-plane.
3. **Sketch Guidance:**
- **Top View:** Circle of radius 3 centered at the origin.
- **Side View:** Visualize one paraboloid opening upwards and the other downwards, meeting at a circle in the \( xy \)-plane.
4. **Triple Integral Setup (without evaluation):**
- The integral will be in the form of:
\[
\int \int \int_{E} \, dV
\]
where \( E \) is the region bounded by the two paraboloids.
- Consider cylindrical coordinates (due to symmetry), with bounds for \( r \) from 0 to 3, \( \theta \) from 0 to \( 2\pi \), and \( z \) from \( r^2 \) to \( 18 - r^2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ba91fa8-c4da-44a3-a037-923d1d08ef86%2F4cd2c697-18fb-4696-8716-ee68838eda7f%2Fdbmjwqk_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 18
**Task:**
Provide a rough sketch of the solid enclosed by the paraboloids \( z = x^2 + y^2 \) and \( z = 18 - x^2 - y^2 \). Use the sketch to set up, but **DO NOT** evaluate, the triple integral that represents the volume between the two surfaces.
**Explanation:**
1. **Paraboloids Description:**
- \( z = x^2 + y^2 \) describes an upward-opening paraboloid with its vertex at the origin \((0,0,0)\).
- \( z = 18 - x^2 - y^2 \) describes a downward-opening paraboloid with its vertex at \((0,0,18)\).
2. **Intersection:**
- To find where the two paraboloids intersect, set their equations equal: \( x^2 + y^2 = 18 - x^2 - y^2 \).
- Simplifying gives \( 2x^2 + 2y^2 = 18 \), or \( x^2 + y^2 = 9 \). This describes a circle of radius 3 in the \( xy \)-plane.
3. **Sketch Guidance:**
- **Top View:** Circle of radius 3 centered at the origin.
- **Side View:** Visualize one paraboloid opening upwards and the other downwards, meeting at a circle in the \( xy \)-plane.
4. **Triple Integral Setup (without evaluation):**
- The integral will be in the form of:
\[
\int \int \int_{E} \, dV
\]
where \( E \) is the region bounded by the two paraboloids.
- Consider cylindrical coordinates (due to symmetry), with bounds for \( r \) from 0 to 3, \( \theta \) from 0 to \( 2\pi \), and \( z \) from \( r^2 \) to \( 18 - r^2 \).
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