Signal Energy
The signal energy expanded from $ t_1\! $ to $ t_2\! $ is defined as $ E = \int_{t_1}^{t_2} \! |x(t)|^2\ dt $.
The following is an example problem to find the signal energy for $ x(t)=e^{2t}\! $ on $ [0,2]\! $:
$ E = \int_{0}^{2} |e^{2t}|^2\ dt \! $
$ = \int_{0}^{2} e^{4t}\ dt \! $
$ = \frac{1}{4}[e^{4t}]_{t=0}^{t=2} \! $
$ = \frac{1}{4}(e^8 -1)\! $
Average Signal Power
The average signal power over an interval $ [t_1,t_2]\! $ is defined as $ P_{avg}=\frac{1}{t_2-t_1} \int_{t_1}^{t_2} |x(t)|^2\ dt \! $.
The following is an example problem to find the average signal power for $ x(t)=e^{2t}\! $ over $ [0,2]\! $:
$ P_{avg}=\frac{1}{2-0} \int_{0}^{2} |e^{2t}|^2\ dt \! $
$ = \frac{1}{2}\int_{0}^{2} e^{4t}\ dt \! $
$ = \frac{1}{2}*\frac{1}{4}[e^{4t}]_{t=0}^{t=2} \! $
$ = \frac{1}{8}(e^8 -1)\! $
