I was talking to somebody today about “CrossFit” and their nonsensical definition of fitness. This is all a shorthand rebuttal to one aspect of the discussion, perhaps I'll discuss further at some other point how nonsensical all their ideas are. Basically, the portion of the idea I am debunking below is their idea that they can quantify what fitness is by plotting an athlete's average power output over various time periods, fitting a curve to that, and then integrating. I discuss, under the presumption that this is even possible (it is not) and valuable (it is not), that this does not work.
I needed to illustrate something, so here's some data to illustrate it.
Let's start with our first point: we've done \( 60 \) kJ of work in \( 1 \) minute, therefore the power is \( 1000 \) W for the first minute.
Sweet.
Now, if you've sustained \( 1000 \) W for an entire minute, then you have achieved that amount for any time domain of less than a minute.
Now, here's the tricky bit, the \( \frac{1}{x} \) bit. Power is work divided by time: \[ P = \frac{W}{T} \] Since the athlete has done \( 60 \) kJ of work in \( 60 \) seconds, that means for any time greater than \( 60 \) seconds, even if the athlete never does any more work, the athlete's power is the following: \[ P(t) = \frac{60kJ}{t \,\, sec} = \frac{60kJ}{t} \] So \( P(t) \propto t^{-1} \) if we never do any further work. So it's a lower bound.
So we have the following graph for the right-hand side:
Having completed \( 60 \) kJ of work, we have completed it for all eternity, which means calculations of average power starting from \( t=0 \) will have this green lower bound. Unfortunately, this integral diverges.
Anyway, to make the complete silly fitness graph, you plot all of your data points and draw the resulting curves in a similar fashion and take the upper envelope as your function. Note it's monotone decreasing - any data points (or any curvy bits) lying below the \( \frac{1}{x} \) curves would be ignored.
Once you have all your points, you'd probably want to do a spline or something on the admissible data points to make a smooth curve.
Anyway, this is all nonsense, because you can't really measure power output, power output is not a reliable way of ranking and comparing athletic performances within one “time domain”, much less across them, and much less across different “modal domains”, so the whole thing is incoherent. And the integral diverges.