Energy
I will calculate the energy expended by the signal $ sin(2t) $ from $ t = 0 $ to $ t = 8\pi $ -
$ E = \int_{0}^{8\pi} \mid sin(2t) \mid^2\, dx $
Integration shows us that:
$ \int_{0}^{8\pi} \mid sin(2t) \mid^2\, dx = t/2-\frac{\sin(2t)\cos(2t)}4 $ evaluated from 0 to 8$ \pi $.
$ E = 4\pi $
Power
I will now calculate the average power of the same function from 0 to 8$ \pi $. Power is very easy to calculate once you have the Energy.
$ P = \frac{1}{8\pi-0}\int_{0}^{8\pi} \mid sin(2t) \mid^2\, dx $
Now for the easy part. Since I already know $ \int_{0}^{8\pi} \mid sin(2t) \mid^2\, dx = 4\pi $ all that's left to do is divide by 8$ \pi $ which yields
$ P = \frac{\pi}{2} $
