Why is a root mean square average used for an AC waveform rather than a plain time average?

1. A sinusoidal current spends half of every cycle positive and half negative. 2. Averaging it directly over a full cycle therefore gives exactly zero. 3. A zero average would wrongly suggest that no energy is delivered, which the glowing lamp disproves. 4. Squaring first removes the sign, so the mean of the squares is not zero. 5. Taking the square root at the end restores the original units, giving a usable effective value. 6. The rms value is smaller than the peak by a factor of the square root of two, so it never equals the peak. _Source: OpenStax College Physics (CC BY 4.0), Ch 20 "Electric Current, Resistance, and Ohm's Law", section Alternating Current_