This question appeared in the GATE exam 2006 ICE paper.
The solution given is, 
I don't see how that the integral is the SD of $f(t)$ even if we assume mean is 0 and it's defined over the entire real line, I think the SD would look like,
$$\left [\int_{-\infty}^{\infty} (t-\mu)^2 f(t)dt \right ]^{\frac{1}{2}}$$
Now of the options only that one looks like a SD, so atleast I hope to understand what standard deviation has to do with "rapidity of change". I thought I understood SD well, it's a measure of dispersion. It tells us how spread out the random variable is.
So please tell me what is a good measure of the rapidity of change in a random variable.
This summarizes the comments above, with perhaps some more info.
The integrals in (a)-(d) are commonly used in signal analysis, where the "signal" $f(t)$ is viewed as either a deterministic or stochastic function of time. None of those integrals can be viewed as the "rapidity of change."
The integral in (a) is the sample path mean; the integral in (b) is the sample path second moment (which can be viewed as the sample path variance if we assume 0 mean); the quantity in (c) is just the square root of that of (b); (d) is a type of auto-correlation: https://en.wikipedia.org/wiki/Autocorrelation
I observe that if we use $f(t)=100$ for all $t$, there is no change in the function over time, so the "rapidity of change" is 0, but none of the integrals in (a)-(d) are 0.
I also observe that the integrals in (a)-(c) are the same if we replace $f(t)$ by $f(\lambda t)$, where $\lambda>0$ is a constant that can speed up or slow down time (i.e., $\lambda$ can directly affect "rapidity of change"). So certainly (a)-(c) have nothing to do with "rapidity of change" as they are not influenced by $\lambda$. If I were forced to pick one of those choices, I would pick (d), since at least varying the $\tau$ parameter can tell you something about how the function changes over a duration of size $\tau$.