Answer
$\frac{25}{96}$
Work Step by Step
Step 1. Given the function $f(x)=\frac{1}{x}$ and the interval $[1,9]$, we can divide the interval into four parts of equal width of $2$ as shown in the figure. And we can find the middle points of each part as $x=2,4,6,8$.
Step 2. We can find the height of each box (figure not to scale) as $f(2)=\frac{1}{2}, f(4)=\frac{1}{4}, f(6)=\frac{1}{6}, f(8)=\frac{1}{8}$
Step 3. We can approximate the area under the function over this interval as the sum of the four rectangles $A=2(f(2)+f(4)+f(6)+f(8))=2(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8})=\frac{25}{12}$
Step 4. The average value of the function over the interval is the area divided by the interval range; thus we have $\bar f_{[1,9]}=\frac{A}{9-1}=\frac{25}{96}$
