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Average value of a function
For a continuous function $f$ on $[a,b]$, its average value is
$$f_{\mathrm{avg}}=\frac{1}{b-a}\int_a^b f(x),dx$$
The integral gives the signed area accumulated across the interval. Dividing by the interval width gives the height of a constant function with the same total accumulation.
Geometrically, imagine redistributing the area under the graph evenly across $[a,b]$. The resulting rectangle has height $f_{\mathrm{avg}}$.
This continuous average also connects integration with rates of change: when $f=F^{\prime}$, the fundamental theorem gives
$$f_{\mathrm{avg}}=\frac{F(b)-F(a)}{b-a}$$
which is the average slope of $F$ across the interval.