7. Sketch the surface 1≤z≤2-x² - y²

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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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 =
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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