What is the average value of the function $F(x) = e ^ -x$ on the interval $[0,4]$ ?

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Answer 1 · 2016-06-21T15:47:15

$1/4 (1 - 1/e^{4})$

graph{e^{-x}x^2 [-10, 10, -5, 5]}

by definition, $y_{ave} = (\int_a^b y \ dx )/(b-a)$

here $y_{ave} = (\int_0^4 e^{-x} \ dx )/(4-0)$

$= (1)/(4) [-e^{-x}]_0^4 = 1/4(-e^{-4} + 1) = 1/4 (1 - 1/e^{4})$

What is the average value of the function F(x) = e ^ -x F(x) = e ^ -x on the interval [0,4][0,4]?