### 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

$P = \frac{\pi}{2}$

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