Compute (1/(b-a))*integral_a^b f(x) dx for linear, quadratic, and sine functions
Select Function Type
f(x)=x+ on [,]
f(x)=x^2+ on [,]
f(x)=sin(x) on [0, pi]
f(x)=sin(x) on [0, pi]
Result
Derivation
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].
Free online calculators and tools covering mathematics, unit conversion, text processing, and daily life. Accurate, fast, mobile-friendly, and completely free to use.