How to Calculate the Average Value of a Function on an Interval [a, b] Using Integration

average value of f(x) on [a, b]

To find the average value of a function f(x) on the interval [a, b], you need to calculate the definite integral of f(x) over the interval [a, b] and divide it by the length of the interval (b – a)

To find the average value of a function f(x) on the interval [a, b], you need to calculate the definite integral of f(x) over the interval [a, b] and divide it by the length of the interval (b – a).

The formula for the average value of f(x) on [a, b] can be written as:

Avg = (1 / (b – a)) * ∫[a to b] f(x) dx

Here, ∫[a to b] represents the definite integral of f(x) with respect to x, evaluated from a to b.

To calculate this, you need to find the antiderivative (or indefinite integral) of f(x), and then evaluate it at the upper bound (b) and lower bound (a) of the interval. Finally, subtract the value of the antiderivative at a from the value at b. This will give you the definite integral of f(x) over the interval [a, b].

Once you’ve obtained the definite integral, divide it by the length of the interval (b – a) to get the average value of f(x) on the interval [a, b].

Note that the average value of a function represents the average height of the function over the given interval. It’s important to remember that the average value might not necessarily be attained by the function at any specific point within the interval; it’s a measure of the overall average height.

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