Average Value of a Function

Calculus 1 · Axiom Academy

LESSON Average Value of a Function Find the single height that, held flat across an interval, traps exactly the same area as a curve — the function's true average. The average of n numbers is their sum over n . To average a function f over [a,b] , replace the sum with the integral (the total accumulated value) and replace n with the interval's length b-a . An ordinary average: total ÷ count The average value of a function on [a,b] 2. The Height That Balances the Area Geometrically, is the height of the rectangle on [a,b] that has the same area as the region under the curve. Raise a flat line to that height and the curve's bulges above it exactly fill the gaps below it. — the exact accumulated value over the interval. — width times the flat average height. If f is continuous, some c in [a,b] has — the curve crosses its own average level. The Mean Value Theorem for Integrals That guaranteed crossing point c is the Mean Value Theorem for integrals : a continuous function attains its average value somewhere on the closed interval — the moment a varying quantity is exactly "typical." 3. Worked Example: f(x)=x^2 on [0,3] Find the average value of f(x)=x^2 over [0,3] . Set a=0 , b=3 , and feed the formula a single integral. Evaluate the integral. The antiderivative of x^2 is , and : Divide by the width b-a = 3 to land the average: square units accumulate from 0 to 3 . Width 3 at height also has area 9 — the parabola's average height is exactly 3 .

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