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9. Find the average distance from the origin to a point between the spheres x² + y² + z² = 1 and x² + y² +z² = 4.

9. Find the average distance from the origin to a point between the spheres x² + y² + z² = 1 and x² + y² +z² = 4.

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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**Problem 9:** Find the average distance from the origin to a point between the spheres \( x^2 + y^2 + z^2 = 1 \) and \( x^2 + y^2 + z^2 = 4 \). **Explanation:** This problem asks us to calculate the average distance from the origin (the point \((0, 0, 0)\)) to points that lie between two concentric spheres. The first sphere is defined by the equation \( x^2 + y^2 + z^2 = 1 \), which is a sphere of radius 1 centered at the origin. The second sphere is defined by \( x^2 + y^2 + z^2 = 4 \), a sphere of radius 2 also centered at the origin. To solve such a problem, one often uses integration techniques over the volume between the spheres, applying spherical coordinates to simplify the problem due to the symmetry around the origin. The solution involves setting up and evaluating an appropriate integral to find the average distance from the origin to the surface between these spheres.
Transcribed Image Text:**Problem 9:** Find the average distance from the origin to a point between the spheres \( x^2 + y^2 + z^2 = 1 \) and \( x^2 + y^2 + z^2 = 4 \). **Explanation:** This problem asks us to calculate the average distance from the origin (the point \((0, 0, 0)\)) to points that lie between two concentric spheres. The first sphere is defined by the equation \( x^2 + y^2 + z^2 = 1 \), which is a sphere of radius 1 centered at the origin. The second sphere is defined by \( x^2 + y^2 + z^2 = 4 \), a sphere of radius 2 also centered at the origin. To solve such a problem, one often uses integration techniques over the volume between the spheres, applying spherical coordinates to simplify the problem due to the symmetry around the origin. The solution involves setting up and evaluating an appropriate integral to find the average distance from the origin to the surface between these spheres.
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