The formula to calculate the Average Function Value over an interval \([a, b]\) is:
\[ \text{Average Value} = \frac{1}{b - a} \int_{a}^{b} f(x) \, dx \]
Where:
The average value of a function over an interval \([a, b]\) is calculated by integrating the function over the interval and then dividing by the length of the interval \((b - a)\). It provides a measure of the function's average height over that interval.
Consider an example where the function is \( f(x) = x^2 \) and the interval is \([0, 1]\):
Using the formula to calculate the Average Function Value:
\[ \text{Average Value} = \frac{1}{1 - 0} \int_{0}^{1} x^2 \, dx = \int_{0}^{1} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{1} = \frac{1}{3} \]
This means that the average value of the function \( f(x) = x^2 \) over the interval [0, 1] is \(\frac{1}{3}\).