Energy and Power
Energy
The following is the energy expended by the signal $ sin(2t) $ from $ t = 0 $ to $ t = 4\pi $:
$ E = \int_{0}^{4\pi} \mid sin(2t) \mid^2\, dx $
Integration tables will tell us that:
$ \int_{0}^{4\pi} \mid sin(2t) \mid^2\, dx = \mid \frac{t}{2} - \frac{sin(2*2*t)}{4*2} \mid $ evaluated at 0 and subtracted from the value at 4$ \pi $.
$ E = 2\pi $
Power
The following is the average power expended by the signal $ sin(2t) $ from $ t = 0 $ to $ t = 4\pi $:
$ P = \frac{1}{4\pi - 0} \int_{0}^{4\pi} \mid sin(2t) \mid^2\, dx $
Since we already know that the integral equals $ 2\pi $, dividing that by $ 4\pi $ will yield the average power.
$ P = \frac{2\pi}{4\pi} = \frac{1}{2} $
