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Function Average Value Calculator

Compute (1/(b-a))*integral_a^b f(x) dx for linear, quadratic, and sine functions

Select Function Type
f(x)=x+ on [,]

Average Value Formula

f_avg = (1/(b-a)) * integral_a^b f(x) dx
Linear: int = (m/2)(b^2-a^2)+b(b-a)
Quadratic: int = (a/3)(b^3-a^3)+c(b-a)
Sine: int = -(cos(b)-cos(a))

The average value of a function on an interval is the constant that would give the same area under the curve. By the Mean Value Theorem for Integrals, a continuous function attains its average value at some point in the interval.

⚠ The average value is NOT the same as the average of the endpoints f(a) and f(b). It uses the integral, which accounts for the entire curve.

What Is Average Value?

The average value of a function generalizes the arithmetic mean to continuous functions. Instead of averaging discrete values, it computes the integral (total area) divided by the interval length. The Mean Value Theorem guarantees the function hits this value somewhere.

Formula

f_avg = (1/(b-a)) * int_a^b f(x)dx. Integrate first, then divide by interval length.

MVT for Integrals

There exists c in [a,b] such that f(c) = f_avg. The function must be continuous. Guarantees the average is achieved.

Geometric Meaning

The rectangle with height f_avg has the same area as the area under f(x) from a to b.

Applications

Average temperature, average velocity (mean value theorem), average current, average signal power in engineering.

Teaching Example: f(x)=2x+1 on [0,4]. Integral = int_0^4 (2x+1)dx = [x^2+x]_0^4 = 16+4=20. Length=4. Avg=20/4=5. Check: MVT: 2c+1=5 -> c=2 in [0,4].

Applications

Physics Engineering Statistics Signal Processing Calculus

Frequently Asked Questions

Average value formula?▼
f_avg = (1/(b-a))*int_a^b f(x)dx. Integrate, then divide by interval length.
MVT for integrals?▼
There exists c in [a,b] where f(c)=f_avg. Guarantees the average value is actually achieved by the function.
Average vs midpoint?▼
Average uses integral (entire curve). Midpoint f((a+b)/2) is just one point. They differ unless f is linear.
Does the theorem always hold?▼
Yes for continuous functions on closed intervals. Discontinuous functions may not achieve their average.

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