Average Value Finder

Math Calculus • Applications of Integrals

Written by STEM Calculators Team Published December 28, 2025 Updated February 24, 2026

4. Average Value Finder

Computes the average value of a function on \([a,b]\): \[ f_{\text{avg}}=\frac{1}{b-a}\int_a^b f(x)\,dx \] and plots the horizontal average line on the graph.

Inputs

Function \(f(x)\)

Use variable x. Constants: pi, e. Supported: + − * / ^, parentheses, sin cos tan, ln log sqrt abs exp. Implicit mult allowed: 2x, (x+1)(x-1).

Lower bound \(a\)

Supports pi, e.

Upper bound \(b\)

Example: \(b=3\) or \(b=\pi\).

Tolerance

Used for numeric integration (adaptive Simpson).

Options

Show grid Shade area \(\int_a^b f(x)\,dx\) Show average line \(y=f_{\text{avg}}\) Show steps Average of \(|f(x)|\) (nonnegative) Try symbolic (when possible)

Presets

Click to load and evaluate.

Ready

Graph

Drag to pan • wheel/pinch to zoom • the average line is the constant height that gives the same area over \([a,b]\).

\(y=f(x)\) \(y=f_{\text{avg}}\) shaded \([a,b]\)

Center \(c\)

Half-width \(w\)

x: 0, y: 0, zoom(px/unit): 60

Result

Enter \(f(x)\) and bounds, then click Evaluate.

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4. Average Value Finder — Theory

Math Calculus • Applications of Integrals

Definition

The average value of a function \(f(x)\) on an interval \([a,b]\) is

\[ f_{\text{avg}}=\frac{1}{b-a}\int_a^b f(x)\,dx. \]

Think of it like the “mean height” of the graph on \([a,b]\), similar to an average temperature over a time period.

Geometric meaning

The integral \(\int_a^b f(x)\,dx\) is the net signed area between the curve and the \(x\)-axis. The average value \(f_{\text{avg}}\) is the height of a rectangle of width \((b-a)\) that has the same area:

\[ (b-a)\,f_{\text{avg}}=\int_a^b f(x)\,dx. \]

On the graph, the calculator draws the horizontal line \(y=f_{\text{avg}}\).

How to compute (FTC steps)

Find an antiderivative \(F\) such that \(F'(x)=f(x)\) (if possible).
Compute the definite integral using the Fundamental Theorem of Calculus:
\[ \int_a^b f(x)\,dx = F(b)-F(a). \]
Divide by the interval length:
\[ f_{\text{avg}}=\frac{F(b)-F(a)}{b-a}. \]

If no simple antiderivative is available, the calculator uses accurate numerical integration (adaptive Simpson).

Example

Find the average value of \(f(x)=x^2\) on \([0,3]\).

\[ f_{\text{avg}}=\frac{1}{3-0}\int_0^3 x^2\,dx =\frac{1}{3}\left[\frac{x^3}{3}\right]_0^3 =\frac{1}{3}\cdot\frac{27}{3} =3. \]

Average of \(\lvert f(x)\rvert\)

Sometimes you want a nonnegative average size (e.g., average speed-like magnitude). Then compute:

\[ f_{\text{avg,abs}}=\frac{1}{b-a}\int_a^b |f(x)|\,dx. \]

This changes the meaning: negative parts contribute positively instead of canceling.

Frequently Asked Questions

What is the average value of a function on an interval?

The average value of f(x) on [a,b] is f_avg = (1/(b-a)) * integral from a to b of f(x) dx. It represents the constant height whose rectangle over width (b-a) has the same area as the integral.

Why does the calculator draw a horizontal average line?

The line y = f_avg shows the mean height of the function over [a,b]. Its rectangle area (b-a) * f_avg matches the value of the definite integral over the same interval.

What does the tolerance setting change?

Tolerance controls the numeric integration accuracy when the integral is evaluated numerically (adaptive Simpson). Smaller tolerance generally increases accuracy but may require more computation.

What is the difference between average of f(x) and average of |f(x)|?

The average of f(x) uses signed area, so positive and negative parts can cancel. The average of |f(x)| makes all contributions nonnegative, giving an average magnitude instead of a signed mean.

When does symbolic integration work instead of numerical integration?

Symbolic integration may work when an antiderivative can be found in a closed form. If a simple antiderivative is not available, the calculator uses numerical integration to approximate the definite integral accurately.