Use a triple integral to find the volume of the solid bounded below by the cone z = √x +y and bounded above by the sphere x² + y² + z² = 18. The volume of the solid is (Type an exact answer.) …... (0,0,√18) x² + y² +2²=18
Use a triple integral to find the volume of the solid bounded below by the cone z = √x +y and bounded above by the sphere x² + y² + z² = 18. The volume of the solid is (Type an exact answer.) …... (0,0,√18) x² + y² +2²=18
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 Statement:
Use a triple integral to find the volume of the solid bounded below by the cone \( z = \sqrt{x^2 + y^2} \) and bounded above by the sphere \( x^2 + y^2 + z^2 = 18 \).
### Diagram Explanation:
In the diagram, there is a 3D representation showing two surfaces:
1. **Cone:** The cone is represented in blue. The equation for the cone is \( z = \sqrt{x^2 + y^2} \). It is open upwards with its vertex at the origin.
2. **Sphere:** The top section of the sphere is depicted in tan. The equation for the sphere is \( x^2 + y^2 + z^2 = 18 \). The center is at the origin with a radius \(\sqrt{18}\).
The intersection of these two surfaces forms the bounds of the solid whose volume needs to be calculated. The point \( (0, 0, \sqrt{18}) \) is marked on the z-axis, indicating the topmost point of the sphere on this axis.
### Solution:
The volume of the solid is calculated using triple integrals. The students are instructed to provide the exact answer in the input box provided below the problem statement:
- "The volume of the solid is [ ] (Type an exact answer.)"
The solution involves evaluating the triple integral using cylindrical coordinates, taking into account the symmetry and bounds provided by the equations of the cone and the sphere.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2c370c8-f220-47ec-b561-9f6a07b2c79a%2Fdf0e2b08-1c45-4f76-8497-1e403aa2bc74%2Fe4xwhpw_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
Use a triple integral to find the volume of the solid bounded below by the cone \( z = \sqrt{x^2 + y^2} \) and bounded above by the sphere \( x^2 + y^2 + z^2 = 18 \).
### Diagram Explanation:
In the diagram, there is a 3D representation showing two surfaces:
1. **Cone:** The cone is represented in blue. The equation for the cone is \( z = \sqrt{x^2 + y^2} \). It is open upwards with its vertex at the origin.
2. **Sphere:** The top section of the sphere is depicted in tan. The equation for the sphere is \( x^2 + y^2 + z^2 = 18 \). The center is at the origin with a radius \(\sqrt{18}\).
The intersection of these two surfaces forms the bounds of the solid whose volume needs to be calculated. The point \( (0, 0, \sqrt{18}) \) is marked on the z-axis, indicating the topmost point of the sphere on this axis.
### Solution:
The volume of the solid is calculated using triple integrals. The students are instructed to provide the exact answer in the input box provided below the problem statement:
- "The volume of the solid is [ ] (Type an exact answer.)"
The solution involves evaluating the triple integral using cylindrical coordinates, taking into account the symmetry and bounds provided by the equations of the cone and the sphere.
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