V Question 2. Let E be the solid region inside the surface x² + y² + x² = 36 and above the surface z = (a) Sketch the solid region E. (b) Sketch the projection of the given solid onto the yz-plane (shade the region). (c) Sketch the projection of the given solid onto the xy-plane (shade the region). 3x² + 3y².
V Question 2. Let E be the solid region inside the surface x² + y² + x² = 36 and above the surface z = (a) Sketch the solid region E. (b) Sketch the projection of the given solid onto the yz-plane (shade the region). (c) Sketch the projection of the given solid onto the xy-plane (shade the region). 3x² + 3y².
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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Transcribed Image Text:### Question 2
Let \( E \) be the solid region inside the surface \( x^2 + y^2 + z^2 = 36 \) and above the surface \( z = \sqrt{3x^2 + 3y^2} \).
#### (a) Sketch the solid region \( E \).
#### (b) Sketch the projection of the given solid onto the \( yz \)-plane (shade the region).
#### (c) Sketch the projection of the given solid onto the \( xy \)-plane (shade the region).
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To solve this problem, we must understand the following:
- The surface \( x^2 + y^2 + z^2 = 36 \) describes a sphere with radius 6 centered at the origin.
- The surface \( z = \sqrt{3x^2 + 3y^2} \) can be written as \( z = \sqrt{3(x^2 + y^2)} \), which describes a cone opening upwards along the z-axis.
#### For part (a):
1. **Visualize the Sphere**:
- The equation \( x^2 + y^2 + z^2 = 36 \) represents a three-dimensional sphere with radius 6 centered at the origin.
2. **Visualize the Cone**:
- The equation \( z = \sqrt{3(x^2 + y^2)} \) represents a cone with its vertex at the origin and opening upwards. The cone is symmetric around the z-axis.
3. **Solid Region \( E \)**:
- The solid region \( E \) is the volume enclosed within the sphere and above the cone.
#### For part (b):
1. **Projection onto the \( yz \)-plane**:
- To find the projection onto the \( yz \)-plane, set \( x = 0 \) in the equations.
- The sphere becomes \( y^2 + z^2 = 36 \) (a circle).
- The cone becomes \( z = \sqrt{3y^2} = \sqrt{3} |y| \).
- Draw the circle \( y^2 + z^2 = 36 \) and shade the region inside of it and above the line \( z = \sqrt{3} |y| \).
#### For part (c):
1. **Projection onto the \( xy \)-plane**
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