7. Sketch the surface 1≤z≤2-x² - y²
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 7:**
**Objective:** Sketch the surface defined by the inequality:
\[ 1 \leq z \leq 2 - x^2 - y^2 \]
**Explanation for the inequality:**
The given inequality describes a region in three-dimensional space (ℝ³).
1. **Lower Bound:** \( z \geq 1 \)
- This represents a horizontal plane at \( z = 1 \).
2. **Upper Bound:** \( z \leq 2 - x^2 - y^2 \)
- This is the equation of an upward-opening paraboloid, which opens downwards because of the minus sign before \( x^2 \) and \( y^2 \).
**Intersection of the bounds:**
- When \( z = 2 - x^2 - y^2 \), substituting \( z = 1 \):
\[ 1 = 2 - x^2 - y^2 \]
\[ x^2 + y^2 = 1 \]
- This represents a circle in the \( x \)-\( y \) plane with a radius of 1 centered at (0,0).
**Three-Dimensional Shape Description:**
- The surfaces to be sketched are between the plane \( z = 1 \) and the paraboloid \( z = 2 - x^2 - y^2 \).
- The region of interest lies within the circular boundary where \( x^2 + y^2 \leq 1 \) and between \( z = 1 \) and the paraboloid.
**Graph's Characteristics and Drawing:**
- **Base:** Circle in the \( x \)-\( y \) plane with radius 1.
- **Curved Upper Surface:** A portion of the paraboloid capped at \( z = 1 \).
The overall region is a "cap" of the paraboloid cut off by the plane \( z = 1 \).
To accurately sketch this surface:
1. Draw the circle \( x^2 + y^2 \leq 1 \) in the \( x \)-\( y \) plane.
2. Represent the plane \( z = 1 \) above this base circle.
3. Show the paraboloid \( z = 2 - x^2 - y^2 \) with the cap created by the intersection at \( z =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F48922dd6-b6f8-4fac-84c6-05a7db5750f6%2Ffffbe0bf-4e84-47c1-9464-8d59efdf69c1%2Fm9jb4s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 7:**
**Objective:** Sketch the surface defined by the inequality:
\[ 1 \leq z \leq 2 - x^2 - y^2 \]
**Explanation for the inequality:**
The given inequality describes a region in three-dimensional space (ℝ³).
1. **Lower Bound:** \( z \geq 1 \)
- This represents a horizontal plane at \( z = 1 \).
2. **Upper Bound:** \( z \leq 2 - x^2 - y^2 \)
- This is the equation of an upward-opening paraboloid, which opens downwards because of the minus sign before \( x^2 \) and \( y^2 \).
**Intersection of the bounds:**
- When \( z = 2 - x^2 - y^2 \), substituting \( z = 1 \):
\[ 1 = 2 - x^2 - y^2 \]
\[ x^2 + y^2 = 1 \]
- This represents a circle in the \( x \)-\( y \) plane with a radius of 1 centered at (0,0).
**Three-Dimensional Shape Description:**
- The surfaces to be sketched are between the plane \( z = 1 \) and the paraboloid \( z = 2 - x^2 - y^2 \).
- The region of interest lies within the circular boundary where \( x^2 + y^2 \leq 1 \) and between \( z = 1 \) and the paraboloid.
**Graph's Characteristics and Drawing:**
- **Base:** Circle in the \( x \)-\( y \) plane with radius 1.
- **Curved Upper Surface:** A portion of the paraboloid capped at \( z = 1 \).
The overall region is a "cap" of the paraboloid cut off by the plane \( z = 1 \).
To accurately sketch this surface:
1. Draw the circle \( x^2 + y^2 \leq 1 \) in the \( x \)-\( y \) plane.
2. Represent the plane \( z = 1 \) above this base circle.
3. Show the paraboloid \( z = 2 - x^2 - y^2 \) with the cap created by the intersection at \( z =
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