What is average value of a function?
The constant height whose rectangle has the same area as ∫_a^b f.
Average Value of a Function Calculator
Find the integral average of f(x)=ax+b or f(x)=ax² on [a,b].
Choose f and interval
Select linear or quadratic monomial, enter coefficients, and the interval endpoints.
About this calculator
How to use Average Value of a Function Calculator, plus the formulas and logic behind the results.
Tool description
The average value of an integrable function on [a,b] is the integral of f divided by the interval length b−a.
How to use
- Select linear px+q or quadratic p x².
- Enter coefficients and endpoints a < b (or a ≠ b).
- Select Calculate to view average value and integral.
- Compare with the midpoint value for linear f.
Formulas & reference
Average = (1/(b−a)) ∫_a^b f(x) dx∫(px+q) dx = (p/2)x² + q x∫ p x² dx = (p/3) x³
Facts
- Linear average equals f at the midpoint
- Units match f’s output units
- Interval length must be nonzero
- Educational exact antiderivatives
Calculation logic
Integrates the selected closed form from a to b, divides by (b−a), and rejects a zero-length interval.
Endpoint order can be reversed; the formula uses b−a signed, so keep a ≠ b.
Average Value of a Function Calculator FAQ
Common questions about this calculator and how to use the results.
Is it the same as the mean of sample points?
It is the continuous mean via integration, not a discrete sample mean.
Why use 1/(b−a)?
It normalizes total area by interval length.
Can a > b?
Yes mathematically; results use signed length b−a.
Why only these f?
Closed antiderivatives keep results exact for this educational tool.
Does quadratic mode include bx+c?
This mode is p x² only; use linear mode for px+q.