Question 4.4.6: Computing the Average Value of a Function Compute the averag...

From (4.3), we have

f_{ ave }=\lim _{n \rightarrow \infty}\left[\frac{1}{b-a} \sum_{i=1}^n f\left(x_i\right) \Delta x\right]=\frac{1}{b-a} \int_a^b f(x) d x (4.3)

f_{\text {ave }}=\frac{1}{\pi-0} \int_0^\pi \sin x d x.

We can approximate the value of this integral by calculating some Riemann sums, to obtain the approximate average, f_{ ave } \approx 0.6366198. (See example 4.4.) In Figure 4.24, we show a graph of y = sin x and its average value on the interval [0, π]. You should note that the two shaded regions have the same area.