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26. Sketch the region in space bounded by z = x² + y² and z=8-x²-y². Also, sketch the projection of the region onto the xy-plane.

26. Sketch the region in space bounded by z = x² + y² and z=8-x²-y². Also, sketch the projection of the region onto the xy-plane.

Advanced Engineering Mathematics
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 26: 3D Region Bounded by Parabolic Surfaces **Task:** Sketch the region in space bounded by the equations: \[ z = x^2 + y^2 \] \[ z = 8 - x^2 - y^2 \] Also, sketch the projection of the region onto the xy-plane. **Explanation:** 1. **Understanding the Surfaces:** - The equation \( z = x^2 + y^2 \) represents a paraboloid opening upwards. - The equation \( z = 8 - x^2 - y^2 \) represents a paraboloid opening downwards, shifted 8 units upwards. 2. **Intersection Curve:** - Set the two equations equal to find the intersection: \[ x^2 + y^2 = 8 - x^2 - y^2 \] \[ 2x^2 + 2y^2 = 8 \] \[ x^2 + y^2 = 4 \] This equation represents a circle with radius 2 in the xy-plane. 3. **Projection onto the xy-plane:** - The projection onto the xy-plane is the circle \( x^2 + y^2 = 4 \). ### Graphical Representation: For an accurate sketch, follow these steps: **3D Sketch:** - Draw the upward-opening paraboloid \( z = x^2 + y^2 \). - Draw the downward-opening paraboloid shifted up by 8 units, \( z = 8 - x^2 - y^2 \). - Highlight the region enclosed between these two surfaces. **2D Projection:** - Draw the circle \( x^2 + y^2 = 4 \) on the xy-plane. By visualizing these steps, you can see how the two parabolic surfaces intersect and the bounded region they form in space. The projection clearly outlines the circular boundary in the xy-plane.
Transcribed Image Text:### Problem 26: 3D Region Bounded by Parabolic Surfaces **Task:** Sketch the region in space bounded by the equations: \[ z = x^2 + y^2 \] \[ z = 8 - x^2 - y^2 \] Also, sketch the projection of the region onto the xy-plane. **Explanation:** 1. **Understanding the Surfaces:** - The equation \( z = x^2 + y^2 \) represents a paraboloid opening upwards. - The equation \( z = 8 - x^2 - y^2 \) represents a paraboloid opening downwards, shifted 8 units upwards. 2. **Intersection Curve:** - Set the two equations equal to find the intersection: \[ x^2 + y^2 = 8 - x^2 - y^2 \] \[ 2x^2 + 2y^2 = 8 \] \[ x^2 + y^2 = 4 \] This equation represents a circle with radius 2 in the xy-plane. 3. **Projection onto the xy-plane:** - The projection onto the xy-plane is the circle \( x^2 + y^2 = 4 \). ### Graphical Representation: For an accurate sketch, follow these steps: **3D Sketch:** - Draw the upward-opening paraboloid \( z = x^2 + y^2 \). - Draw the downward-opening paraboloid shifted up by 8 units, \( z = 8 - x^2 - y^2 \). - Highlight the region enclosed between these two surfaces. **2D Projection:** - Draw the circle \( x^2 + y^2 = 4 \) on the xy-plane. By visualizing these steps, you can see how the two parabolic surfaces intersect and the bounded region they form in space. The projection clearly outlines the circular boundary in the xy-plane.
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