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Mean of a Function

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].

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Mean of a Function -

Tag(s) : Functions

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Mean of a Function

Average of a Function Calculator

 Result Format Automatic Selection Exact Value (when possible) Approximate Numerical Value Scientific Notation

What is the average value of a function? (Definition)

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 $[a,b]$

How to compute the average value of a function?

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 $$\bar{f} = \frac{1}{1-0} \int^1_0 f(x) \mathrm{ dx} = F(1)-F(0) = \frac{1}{2}$$

How to compute the average value of a function with dCode?

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

How to calculate the primitive of a function?

The primitive calculator is a also tool proposed on dCode.

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Mean of a Function on dCode.fr [online website], retrieved on 2024-07-18, https://www.dcode.fr/mean-function

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