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Use a 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 = 162. (0,0,√162) x² + y² +2²=162

Use a 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 = 162. (0,0,√162) x² + y² +2²=162

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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How do I acquire the bounds for the triple integral?

**Volume of a Solid Bounded by a Cone and a Sphere** To find the volume of the solid bounded below by the cone \( z = \sqrt{x^2 + y^2} \) and above by the sphere \( x^2 + y^2 + z^2 = 162 \), we use a triple integral. **Graphical Explanation:** The diagram displays a three-dimensional coordinate system with the \( x \)-, \( y \)-, and \( z \)-axes. 1. **Cone:** - The cone is represented by the equation \( z = \sqrt{x^2 + y^2} \). - It is shown as a blue, inverted conical shape with its apex at the origin \((0,0,0)\) and extending upwards. 2. **Sphere:** - The sphere in the diagram is described by the equation \( x^2 + y^2 + z^2 = 162 \). - It is depicted as a yellowish cap that sits atop the conical portion. 3. **Intersection:** - The two surfaces intersect at a circle in the \( xy \)-plane. - At the top of the sphere, a point labeled \((0,0,\sqrt{162})\) indicates the z-coordinate of the uppermost part of the sphere. By setting up a triple integral in cylindrical or spherical coordinates, we can calculate the volume of the solid region that lies within these bounds.
Transcribed Image Text:**Volume of a Solid Bounded by a Cone and a Sphere** To find the volume of the solid bounded below by the cone \( z = \sqrt{x^2 + y^2} \) and above by the sphere \( x^2 + y^2 + z^2 = 162 \), we use a triple integral. **Graphical Explanation:** The diagram displays a three-dimensional coordinate system with the \( x \)-, \( y \)-, and \( z \)-axes. 1. **Cone:** - The cone is represented by the equation \( z = \sqrt{x^2 + y^2} \). - It is shown as a blue, inverted conical shape with its apex at the origin \((0,0,0)\) and extending upwards. 2. **Sphere:** - The sphere in the diagram is described by the equation \( x^2 + y^2 + z^2 = 162 \). - It is depicted as a yellowish cap that sits atop the conical portion. 3. **Intersection:** - The two surfaces intersect at a circle in the \( xy \)-plane. - At the top of the sphere, a point labeled \((0,0,\sqrt{162})\) indicates the z-coordinate of the uppermost part of the sphere. By setting up a triple integral in cylindrical or spherical coordinates, we can calculate the volume of the solid region that lies within these bounds.
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