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

### The average value of a function f(x) on the interval [a, b] can be found using the mean value theorem for integrals

The average value of a function f(x) on the interval [a, b] can be found using the mean value theorem for integrals. The formula to calculate the average value is given by:

\[ \text{Average value of } f(x) = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx \]

To compute the average value, follow these steps:

1. Find the definite integral of f(x) over the interval [a, b]. This represents the total accumulation of the function over the given interval.

2. Subtract the integral value at the lower bound (a) from the value at the upper bound (b): \[ \int_{a}^{b} f(x) \, dx \]

3. Divide the result by the length of the interval (b – a): \(\frac{1}{b-a}\)

The final outcome will provide the average value of the function f(x) on the interval [a, b]. This average value represents the height of a horizontal line that, if the graph of f(x) is perfectly balanced, would give the same area under the curve as the original function over the same interval.

It’s important to note that the average value of a function can differ significantly from the function’s actual value at certain points within the interval.

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