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Average Value of a Function Online Calculator

Average value of a function calculator. Enter f(x) on [a,b] to compute the mean f̄=(1/(b−a))∫f dx via the mean-value theorem—ideal for integral chapters.

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f̄ = (1/(b−a)) ∫ₐᵇ f(x) dx
Supports + - * / ^ ( ) and sin cos tan sec cot sqrt exp log abs asin atan pi e; write 2*x, not 2x
Lower limit of integration
Upper limit of integration

📖 Tutorial | Average Value of a Function

1. Definition
The average value of f on [a,b] is f̄=(1/(b−a))∫ₐᵇ f(x)dx. The mean-value theorem guarantees a c∈[a,b] with f(c)=f̄.
2. Symbols
SymbolsMeaning
f̄Average height
∫f dxArea under the curve
(b−a)Length of interval
cPoint with f(c)=f̄
3. How it works
  • Integrate ∫ₐᵇ f dx numerically;
  • Divide by the length (b−a);
  • This flattens the area into an equal rectangle.
4. Steps
  1. Enter f(x);
  2. Enter [a,b];
  3. Click Calculate for f̄.
5. Example
Example: f(x)=x² on [0,1]: ∫=1/3, length 1, so f̄=1/3≈0.333333.
6. Pitfalls
The average is integral/length, not the arithmetic mean of endpoints;
f̄ is a height of an equal-area rectangle;
The point c is generally not the midpoint.

❓ FAQ | Average Value of a Function

Like an ordinary mean?
It is a continuous mean: infinitely dense averaging, equal to integral/length.
Why divide by (b−a)?
The integral is the “sum”; dividing by length gives the mean.
Is there a c with f(c)=f̄?
For continuous functions, the mean-value theorem guarantees it.
Average of sine on [0,π]?
2/π≈0.6366.
Which functions?
+−*/^ and common elementary functions.