Average Value of Function Calculator

Compute the average value of any function over an interval using definite integration. Adjustable precision, symbolic results + numeric approximation with interactive graph.

Average Value Formula: \( \bar{f} = \frac{1}{b-a} \int_a^b f(x) \, dx \)

Where \( \int_a^b f(x) \, dx \) is the definite integral over the interval.

Use ^ for exponent, * for multiplication, sin(), cos(), exp(), log(), sqrt(), etc.
x² on [0,2]
sin(x) on [0,π]
eˣ on [0,1]
√x on [1,4]
1/x on [1,e]
3x²+2x-5 on [-1,2]
Number of subintervals for numerical integration.
Computing integral & average...

Understanding Average Value of a Function

The average value of a function \(f(x)\) over the interval \([a,b]\) is defined by the definite integral. It generalizes the concept of an average of a finite set of numbers to a continuum.

Mean Value Theorem for Integrals:

If f is continuous on [a,b], there exists at least one number c in [a,b] such that:

\( f(c) = \frac{1}{b-a} \int_a^b f(x) \, dx \)

i.e., the function attains its average value at some point in the interval.

Common Integrals (for reference)

Function f(x) Indefinite Integral ∫ f(x) dx Example
\(x^n\) (n ≠ -1) \(\frac{x^{n+1}}{n+1} + C\) ∫ x² dx = x³/3
\(1/x\) \(\ln|x| + C\) ∫ 1/x dx = ln|x|
\(e^x\) \(e^x + C\) ∫ eˣ dx = eˣ
\(\sin x\) \(-\cos x + C\) ∫ sin x dx = -cos x
\(\cos x\) \(\sin x + C\) ∫ cos x dx = sin x
\(\sec^2 x\) \(\tan x + C\) ∫ sec² dx = tan x
\(\frac{1}{\sqrt{1-x^2}}\) \(\arcsin x + C\)
\(\frac{1}{1+x^2}\) \(\arctan x + C\)

Geometric Interpretation

1

Area equivalence: The area under the curve over [a,b] equals the area of a rectangle of width (b-a) and height equal to the average value.

2

Mean Value Theorem: For continuous functions, the horizontal line at the average value will always intersect the curve at least once.

3

Connection to averages: If you sample the function at infinitely many points, the arithmetic mean converges to the average value.

Applications

  • Physics: Average velocity, average current, average force over a distance
  • Economics: Average cost, average revenue over a production level
  • Probability: Expected value of a continuous random variable
  • Engineering: Mean temperature over a time interval, average signal power

Calculator Features:

  • Symbolic integration via nerdamer.js for exact results when possible
  • Adjustable numerical precision (low/medium/high) for adaptive accuracy
  • Clear indication when numerical approximation is used (info icon)
  • Interactive graph showing function, average line, and interval
  • Supports trigonometric, exponential, logarithmic, and polynomial functions

Frequently Asked Questions

It is the constant value that, if maintained over the interval, would yield the same total accumulation (area) as the actual function. Graphically, it's the height of a rectangle with base [a,b] whose area equals the area under the curve.

The definition works for any integrable function (piecewise continuous, with finite discontinuities). However, the Mean Value Theorem for integrals requires continuity to guarantee the function attains the average value.

It uses nerdamer.js to compute the antiderivative symbolically. If the antiderivative is found and can be evaluated at bounds without free variables, an exact result is shown. Otherwise, it falls back to an accurate numerical integration (adaptive Simpson) and provides a decimal approximation. The info icon indicates when numerical approximation is used.

The integral may still exist as an improper integral. Our numerical method may fail or give inaccurate results. For singularities, ensure the interval avoids them or use limits. The calculator may warn you if evaluation fails.

Yes! Use the "Numerical Precision" dropdown to choose low (fast), medium (recommended), or high (more accurate) settings. This controls the number of subintervals in the numerical integration routine.