53–54 The average value of a function f ( x , y , z ) over a solid region E is defined to be f a v e = 1 v ( E ) ∭ E f ( x , y , z ) d V where V ( E ) is the volume of E. For instance, if ρ is a density function, then ρ a v e is the average density of E . Find the average height of the points in the solid hemisphere x 2 + y 2 + z 2 ≤ 1 , z ≥ 0 .
53–54 The average value of a function f ( x , y , z ) over a solid region E is defined to be f a v e = 1 v ( E ) ∭ E f ( x , y , z ) d V where V ( E ) is the volume of E. For instance, if ρ is a density function, then ρ a v e is the average density of E . Find the average height of the points in the solid hemisphere x 2 + y 2 + z 2 ≤ 1 , z ≥ 0 .
Solution Summary: The author calculates the average height of the points in the solid hemisphere by finding its z coordinate.
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