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

Question

Ezra Adams · Accepted Answer

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

plot{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})#

Christopher Allers · Answer

undefined To find the average value of a function $ F(x) $ on the interval $[a, b]$, you use the formula:

$$ \text{Average Value} = \frac{1}{b - a} \int_{a}^{b} F(x) \, dx $$

For $ F(x) = e^{-x} $ on the interval $[0, 4]$:

$$ \text{Average Value} = \frac{1}{4 - 0} \int_{0}^{4} e^{-x} \, dx $$

$$ = \frac{1}{4} \left[-e^{-x}\right]_{0}^{4} $$

$$ = \frac{1}{4} \left(-e^{-4} + e^0\right) $$

$$ = \frac{1}{4} \left(-\frac{1}{e^4} + 1\right) $$

$$ \approx \frac{1 - e^{-4}}{4} $$

$$ \approx \frac{1 - \frac{1}{e^4}}{4} $$

$$ \approx \frac{e^4 - 1}{4e^4} $$

$$ \approx 0.3293 $$

So, the average value of $ F(x) = e^{-x} $ on the interval $[0, 4]$ is approximately $ 0.3293 $.