Tool to compute the mean of a function (continuous) in order to find the average value of its integral over a given interval [a,b].

Mean of a Function - dCode

Tag(s) : Functions

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The mean of a function $ f $ is noted $ \bar{f} $ and is calculated over an interval $ [a,b] $ with the formula $$ \bar{f} = \frac{1}{b-a} \int_a^b f(x) \mathrm{ dx} $$ with $$ \int^b_a f(x) \mathrm{ dx} = F(b)-F(a) $$ with $ F(x) $ the primitive of $ f(x) $ over the interval $ [a,b] $

Calculate the integral of the function over the desired interval and divide by the size of the interval.

__Example:__ Calculate the mean of the function $ f(x) = x $ over the interval $ [0;1] $ necessitate to calculate the primitive $ F(x) = \frac{1}{2} x^2 $ and then $$ \bar{f} = \frac{1}{1-0} \int^1_0 f(x) \mathrm{ dx} = F(1)-F(0) = \frac{1}{2} $$

Indicate the function with lower and upper bounds (that delimitate the interval) and the variable to integrate with.

Symbolic values are allowed and included in the calculation.

dCode will compute the integral of the function and will find the average value

The function must be continuous and convergent to obtain results

The primitive calculator is a also tool proposed on dCode.

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Cite as source (bibliography):

*Mean of a Function* on dCode.fr [online website], retrieved on 2024-07-18,

mean,average,function,integral,integrate,primitive

https://www.dcode.fr/mean-function

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