Use triple integral to find the volume of the solid bounded below by the cone z = √√x² + y² and bounded above by the sphere x² + y² + z² = 98. The volume of the solid is (Type an exact answer.) 27 3 3 981 49 -343, cubic units. (0,0,√98) x² + y² +2²=98 z=√x² + y²
Use triple integral to find the volume of the solid bounded below by the cone z = √√x² + y² and bounded above by the sphere x² + y² + z² = 98. The volume of the solid is (Type an exact answer.) 27 3 3 981 49 -343, cubic units. (0,0,√98) x² + y² +2²=98 z=√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 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 \( x^2 + y^2 + z^2 = 98 \).
### Diagram Explanation
The diagram on the right shows a 3D visualization of the solid. It consists of a cone and a sphere:
- **Cone:** The base of the cone is in the \(xy\)-plane, with the apex at the origin. The cone opens upwards with the equation \( z = \sqrt{x^2 + y^2} \).
- **Sphere:** The sphere is centered at the origin with the equation \( x^2 + y^2 + z^2 = 98 \). The part of the sphere above the cone forms the upper boundary of the solid.
### Calculation and Answer
The volume of the solid is calculated using a triple integral. The result is given as:
\[
\frac{2\pi}{3} \left( \frac{3}{2}98 - 49\sqrt{98} - 343 \right) \text{ cubic units}.
\]
This is the exact volume of the solid.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2c370c8-f220-47ec-b561-9f6a07b2c79a%2Fcab31e61-8d54-4af3-a999-a1bc8b8a6a5b%2F8o98sfk_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 \( x^2 + y^2 + z^2 = 98 \).
### Diagram Explanation
The diagram on the right shows a 3D visualization of the solid. It consists of a cone and a sphere:
- **Cone:** The base of the cone is in the \(xy\)-plane, with the apex at the origin. The cone opens upwards with the equation \( z = \sqrt{x^2 + y^2} \).
- **Sphere:** The sphere is centered at the origin with the equation \( x^2 + y^2 + z^2 = 98 \). The part of the sphere above the cone forms the upper boundary of the solid.
### Calculation and Answer
The volume of the solid is calculated using a triple integral. The result is given as:
\[
\frac{2\pi}{3} \left( \frac{3}{2}98 - 49\sqrt{98} - 343 \right) \text{ cubic units}.
\]
This is the exact volume of the solid.
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