Signal
We will compute the Power and Energy of a 440HZ sin wave, also known as an "A".
$ x(t)=sin(2\pi440t)\! $.
Average Power
Average power of the 440 Hz sine wave.
$ E=\frac{1}{2\pi-0}\int_0^{2\pi}{|sin(2\pi440t)|^2dt} $
$ =\frac{1}{2\pi-0}\frac{1}{2}\int_0^{2\pi}(|1-cos(4\pi440t)|)dt $
$ =\frac{1}{4\pi}(t-\frac{1}{4\pi440}sin(4\pi440t))|_{t=0}^{t=2\pi} $
$ =\frac{1}{4\pi}(2\pi-(1.7305e^-4)-0-0) $
$ =\frac{1}{2} $
Energy
$ P = \int_0^2\! |\sin(2\pi440t)|^2\ dt $
$ P = {1\over 2}\int_0^2\! |1-\cos(4\pi440t)| dt $
$ P = {1\over 2} ( t - {1\over 4\pi440}\sin(4\pi440t) )\mid_0^{2\pi} $
$ P = {1\over 2}(2\pi) - {1\over 8\pi440}\sin(4\pi440*2\pi) - [{1\over 2}(0) - {1\over 8\pi440}\sin(4\pi440*0)] $
$ P = \pi - {1\over8\pi440}\sin(8\pi^2440) $
$ P = 3.1415\! $
