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.
A 20 foot ladder rests on level ground; its head (top) is against a vertical wall. The bottom of the ladder begins by being 12 feet from the wall but begins moving away at the rate of 0.1 feet per second. At what rate is the top of the ladder slipping down the wall? You may use a calculator.
Explain the focus and reasons for establishment of 12.4.1(root test) and 12.4.2(ratio test)
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