Definition of average power

I am slightly confused by the definition of average power if the power function $p(t)$ is sinusoidal. Why is it that only one period is considered?

I mean I know that it simplifies calculations but if we assume that the period of $p(t)$ is $T$ and I compute the average power over $[0,\sqrt{2}T]$, I do not get the same result had I computed the average power over $[0,T].$

Case in point: If $p(t)=\frac{1}{2}(1+\cos(2x))$ then the average power is not the same for both cases mentioned...

sophiecentaur