Applications: physics and signals

The average value is not a theoretical curiosity: it is an everyday tool in physics and signal processing, wherever a quantity varies continuously.

Average speed. If the speed v(t) of a moving object varies over time between instants t₁ and t₂, its average speed is the average value of v:

   v_avg = 1/(t₂ − t₁) · ∫ from t₁ to t₂ v(t) dt = (distance traveled) / (time)

We recover the elementary definition "distance / time", because the integral of speed is precisely the distance. The integral formula generalizes it to any kind of variation.

Average value of a signal. In electricity, an alternating voltage u(t) varies constantly. Its average value over one period tells whether a "DC" component remains. For a centered sinusoidal signal, this average is zero — the positive and negative half-cycles cancel out.

   centered alternating signal  ->  average value = 0 (over one period)

That is why these signals are often characterized not by their average (zero) but by the average of their square (whose square root is the effective value, or RMS), which is not zero and measures the "power" of the signal.

   average value      : 1/(b−a) ∫ f                  (DC component)
   effective value    : square root of the mean of f²  (signal power)

The common thread: whenever a quantity varies continuously and we want a single representative value, it is the average value — hence an integral — that we compute. Speed, flow rate, temperature, voltage: the same idea everywhere.