Resolved: "What is the average value of the function..."

Karla Thompson

Answered question

2022-04-17

What is the average value of the function

F (x) = x e^{x^{2}}

on the interval [0,2]?

shulsellsmine2aaf

Beginner2022-04-18Added 5 answers

The average value of a continuous function F over an interval [a,b] is

\frac{1}{b - a} \int_{a}^{b} F (x) d x

.
For

F (x) = x e^{x^{2}}

over the interval [0,2], this becomes

\frac{1}{2} \int_{0}^{2} x e^{x^{2}} d x

The integral can be done using the substitution

u = x^{2}

, making du=2x dx. Also, when x=0, u=0 and when x=2,

u = 2^{2} = 4

. Therefore, the quantity above can be written as

\frac{1}{2} \int_{0}^{4} \frac{1}{2} e^{u} d u = {[\frac{1}{4} e^{u}]}_{0}^{4} = \frac{1}{4} e^{4} - \frac{1}{4} e^{0} = \frac{1}{4} e^{4} - \frac{1}{4} \approx 13.3995

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