Given the Signal x(t) = 4sin(2 * pi * 6t), Find the energy and power of the signal from 2 to 6 seconds.
Energy
$ E=\int_2^6 |x(t)|^2 dt $
$ E=\int_2^6 |4sin(12\pi t)|^2 dt $
$ E=16*\int_2^6 sin(12\pi t)^2 dt $
$ E=16*\int_2^6 sin(12\pi t)^2 dt $
$ E=16*(\dfrac{12 \pi t}{2}-\dfrac{sin(2*(12\pi t))}{4})\mid_2^6 $
$ E=16*(6\pi t-\dfrac{sin(24\pi t)}{4})\mid_2^6 $
$ E=96\pi *t-\dfrac{16sin(24\pi *t)}{4}\mid_2^6 $
$ E=96\pi *6-\dfrac{16sin(24\pi *5)}{4}-(96\pi *2-\dfrac{16sin(24\pi *1)}{4} $
$ E=384\pi $
Power
$ P=\dfrac{1}{t2-t1}\int_2^6 |x(t)|^2 dt $
$ P=\dfrac{1}{6-2}\int_2^6 |4sin(12\pi t)|^2 dt $
$ P=\dfrac{16}{4}*\int_2^6 sin(12\pi t)^2 dt $
$ P=\dfrac{16}{4}*\int_2^6 sin(12\pi t)^2 dt $
$ P=\dfrac{16}{4}*(\dfrac{12 \pi t}{2}-\dfrac{sin(2*(12\pi t))}{4})\mid_2^6 $
$ P=\dfrac{16}{4}*(6\pi t-\dfrac{sin(24\pi t)}{4})\mid_2^6 $
$ P=\dfrac{96}{4}\pi *t-\dfrac{16sin(24\pi *t)}{16}\mid_2^6 $
$ P=\dfrac{96}{4}\pi *6-\dfrac{16sin(24\pi *6)}{16}-(\dfrac{96}{4}\pi *2-\dfrac{16sin(24\pi *2)}{16} $
Power =96 *pi
