Q:

What's the point of an RMS value?

the RMS (root mean square) value of $f(x)$ is defined as:
$$f(x)_{rms}=\sqrt{\frac{\int^b_a (f(x))^2dx}{b-a}}$$
Why do we do this very specific thing of taking the square, the mean, and then the square root of the function? For an AC circuit, why does this tell us the power consumption and not something like the expression below?
$$\frac{\int^b_a|f(x)|dx}{b-a}$$

A:

Consider the instantaneous current $i(t)$ through a resistance $R$. The instantaneous power dissipation is
$$P(t) = Ri^2(t).$$
The average power dissipation during a sufficiently long time $T$ after $t=0$ is
$$P_{avg}=\frac{1}{T}\int\limits_0^T{P(t)dt}=R\frac{1}{T}\int\limits_0^T{i^2(t)dt}=R\sqrt{\frac{1}{T}\int\limits_0^T{i^2(t)dt}}^2=Ri_{rms}^2.$$
This illustrates why RMS values are useful: you can use them to calculate the average power from e.g. the RMS voltage and current in the exact same way you would use the instantaneous values to calculate the instantaneous power (in AC you do need to take phase angles into account). Other ways of averaging quantities, like the mean absolute value as you propose, do not have this property.

A:

By definition the rms,  also called the effective or heating value of AC, is the equivalent of DC with respect to resistance heating. The reason for taking the square is because both positive and negative values of current equally produce resistance  heating.
Hope this helps.

