You may not have a half period to integrate over. A night light may use a single diode that only draws current over half a period and no current on the other half period.

But the big thing to remember is that the vector computations are only valid when current and voltage are sine waves. Real-world non-linear loads will not have a current curve that is rotated in phase relative to the voltage. The current may be perfectly in phase, but be just short spikes. This is typical of a system with a rectifier followed by a capacitor.

That is why you measure the Irms, Vrms and Prms individually. And for computing Prms, you do the integration by accumulating the products of the instantaneous I and V values.

And you do not integrate for a half period, but you integrate for many full periods, since real-life measurements of real-life signals will contain a lot of noise. There is not "a correct" value, but a value that is constantly changing with a distribution around an average.