What is the average value of the function f(x)=x2+4x2 on the interval [3,5]?

The average value of the function f on the interval [a,b] is equal to:1b−a∫abf(x)dxHere this gives us an average value of:15−3∫35x2+4x2dx=12∫35(1+4x−2)dxUsing ∫dx=x and ∫xndx=xn+1n+1 for n≠−1:=12[x−4x]35=12(5−45)−12(3−43)=1915

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High School
Calculus and Analysis
Calculus 1
Derivatives

Emilia Hoffman

Answered question

2022-04-16

What is the average value of the function

f (x) = \frac{x^{2} + 4}{x^{2}}

on the interval [3,5]?

Answer & Explanation

Alan Bean

Beginner2022-04-17Added 10 answers

The average value of the function f on the interval [a,b] is equal to:

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

Here this gives us an average value of:

\frac{1}{5 - 3} \int_{3}^{5} \frac{x^{2} + 4}{x^{2}} d x = \frac{1}{2} \int_{3}^{5} (1 + 4 x^{- 2}) d x

Using

\int d x = x

and

\int x^{n} d x = \frac{x^{n + 1}}{n + 1}

for

n \neq - 1 :

= \frac{1}{2} {[x - \frac{4}{x}]}_{3}^{5} = \frac{1}{2} (5 - \frac{4}{5}) - \frac{1}{2} (3 - \frac{4}{3}) = \frac{19}{15}

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