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7. Set up and evaluate a spherical coordinates integral for the volume of the solid bounded below by the sphere p = 2 cos (which is the sphere x² + y² + (z − 1)² = 1) and above by the cone z = √√√x² + y². p = 2 cos o y

7. Set up and evaluate a spherical coordinates integral for the volume of the solid bounded below by the sphere p = 2 cos (which is the sphere x² + y² + (z − 1)² = 1) and above by the cone z = √√√x² + y². p = 2 cos o y

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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Use spherical Coordinates 

**Problem 7:** Set up and evaluate a spherical coordinates integral for the volume of the solid bounded below by the sphere \(\rho = 2 \cos \varphi\) (which is the sphere \(x^2 + y^2 + (z-1)^2 = 1\)) and above by the cone \(z = \sqrt{x^2 + y^2}\). **Diagram Explanation:** The diagram illustrates a 3D coordinate system marked with \(x\), \(y\), and \(z\) axes. Within it, there is a solid bounded below by a sphere described by the equation \(\rho = 2 \cos \varphi\) and above by a cone given by \(z = \sqrt{x^2 + y^2}\). - The sphere is centered at \((0, 0, 1)\) with a radius of 1, as derived from rearranging the sphere equation \(x^2 + y^2 + (z-1)^2 = 1\). - The cone opens upwards from the origin with its vertex at the origin and follows the equation \(z = \sqrt{x^2 + y^2}\), which describes a right circular cone. The task is to find the volume of this region using an integral in spherical coordinates.
Transcribed Image Text:**Problem 7:** Set up and evaluate a spherical coordinates integral for the volume of the solid bounded below by the sphere \(\rho = 2 \cos \varphi\) (which is the sphere \(x^2 + y^2 + (z-1)^2 = 1\)) and above by the cone \(z = \sqrt{x^2 + y^2}\). **Diagram Explanation:** The diagram illustrates a 3D coordinate system marked with \(x\), \(y\), and \(z\) axes. Within it, there is a solid bounded below by a sphere described by the equation \(\rho = 2 \cos \varphi\) and above by a cone given by \(z = \sqrt{x^2 + y^2}\). - The sphere is centered at \((0, 0, 1)\) with a radius of 1, as derived from rearranging the sphere equation \(x^2 + y^2 + (z-1)^2 = 1\). - The cone opens upwards from the origin with its vertex at the origin and follows the equation \(z = \sqrt{x^2 + y^2}\), which describes a right circular cone. The task is to find the volume of this region using an integral in spherical coordinates.
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