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To find the average value of a function f(x, y, z) on a solid region E, we have the formula 1 avg(f) /// f(x, y, z) dV volume(E) E (a) Find a function f(x, y, z) which gives the distance from a point (x, y, z) to the origin (0,0,0). (b) Suppose E is a solid ball of radius 2 centered at the origin. Use the formula above to find the average value of f (from part (a)) on the E. Hint: the volume of a sphere of radius r is V = (c) Is the average distance from a point in E to the origin more or less than half the radius of E?

To find the average value of a function f(x, y, z) on a solid region E, we have the formula 1 avg(f) /// f(x, y, z) dV volume(E) E (a) Find a function f(x, y, z) which gives the distance from a point (x, y, z) to the origin (0,0,0). (b) Suppose E is a solid ball of radius 2 centered at the origin. Use the formula above to find the average value of f (from part (a)) on the E. Hint: the volume of a sphere of radius r is V = (c) Is the average distance from a point in E to the origin more or less than half the radius of E?

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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To find the average value of a function f(x, y, z) on a solid region E, we have the formula 1 avg(f) f(2, y, z) dV volume(E) (a) Find a function f(x, y, z) which gives the distance from a point (x, y, z) to the origin (0,0, 0). (b) Suppose E is a solid ball of radius 2 centered at the origin. Use the formula above to find the average value of f (from part (a)) on the E. Hint: the volume of a sphere of radius r is V = 4r. (c) Is the average distance from a point in E to the origin more or less than half the radius of E?
Transcribed Image Text:To find the average value of a function f(x, y, z) on a solid region E, we have the formula 1 avg(f) f(2, y, z) dV volume(E) (a) Find a function f(x, y, z) which gives the distance from a point (x, y, z) to the origin (0,0, 0). (b) Suppose E is a solid ball of radius 2 centered at the origin. Use the formula above to find the average value of f (from part (a)) on the E. Hint: the volume of a sphere of radius r is V = 4r. (c) Is the average distance from a point in E to the origin more or less than half the radius of E?
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