Average Value of a Function
Given a continuous function fx, its average value over [a , b] is defined as 1b − a∫ab fx ⅆx.
Suppose that you are driving a car in a straight line and that v(t) is your velocity at time, t. The distance you traveled over the time interval from t = a to t = b can be written using the definite integral ∫abvt ⅆt. Therefore, your average driving speed was 1b − a∫ab vt ⅆt, which represents the average value of the velocity function.
A geometric way to interpret the average value of a function is in terms of area. According to the Mean Value Theorem for integrals, given a continuous function f(x) defined over the closed interval a, b, there exists a point t' in the interior a, b such that: ft', the instantaneous value of f at t' is equal to the average value of fx over the whole interval a, b. Thus, multiplying by b−a on both sides equals:
ft'b − a = ∫abft ⅆt .
In other words, the area is defined by a rectangle, whose height is the average value of a function and whose width is an interval equal to the area under the entire function, occurring over the same interval.
Choose a function and interval over which to plot the function and the rectangle showing its average value. Then, adjust the slider to see how changing the interval affects the area under the rectangle. Verify that the area under the rectangle is the same as the area under the entire curve.
f(x) Example 1Example 2Example 3Custom
a = , b =
∫abft ⅆt =
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