Chapter 4 - Integration - 4.5 Exercises - Page 303: 84

$$\eqalign{ & \left( {\text{a}} \right){I_{avg}} = \frac{4}{\pi }{\text{Amperes}} \cr & \left( {\text{b}} \right){I_{avg}} = \frac{{5 - 2\sqrt 2 }}{{2\pi }}{\text{Amperes}} \cr & \left( {\text{c}} \right){I_{avg}} = 0{\text{Amperes}} \cr} $$

$$\eqalign{ & {\text{The oscillating current }}\left( {{\text{in amperes}}} \right){\text{in an electrical circuit is}} \cr & I = 2\sin \left( {60\pi t} \right) + \cos \left( {120\pi t} \right),{\text{ }}t{\text{ in seconds}} \cr & {\text{The average value is:}} \cr & {I_{avg}} = \frac{1}{{b - a}}\int_a^b {\left[ {2\sin \left( {60\pi t} \right) + \cos \left( {120\pi t} \right)} \right]dt} \cr & {I_{avg}} = \frac{1}{{b - a}}\left[ { - \frac{2}{{60\pi }}\cos \left( {60\pi t} \right) + \frac{1}{{120\pi }}\sin \left( {120\pi t} \right)} \right]_a^b \cr & {I_{avg}} = \frac{1}{{b - a}}\left[ { - \frac{{\cos \left( {60\pi t} \right)}}{{30\pi }} + \frac{{\sin \left( {120\pi t} \right)}}{{120\pi }}} \right]_a^b \cr & \cr & \left( {\text{a}} \right){\text{ }}0 \leqslant t \leqslant \frac{1}{{60}} \cr & {I_{avg}} = \frac{1}{{1/60 - 0}}\left[ { - \frac{{\cos \left( {60\pi t} \right)}}{{30\pi }} + \frac{{\sin \left( {120\pi t} \right)}}{{120\pi }}} \right]_0^{1/60} \cr & {I_{avg}} = 60\left[ { - \frac{{\cos \left( {60\left( {1/60} \right)\pi } \right)}}{{30\pi }} + \frac{{\sin \left( {120\left( {1/60} \right)\pi } \right)}}{{120\pi }}} \right]_0^{1/60} \cr & {I_{avg}} = 60\left[ { - \frac{{\cos \left( \pi \right)}}{{30\pi }} + \frac{{\sin \left( {2\pi } \right)}}{{120\pi }}} \right] - 60\left[ { - \frac{{\cos \left( 0 \right)}}{{30\pi }}} \right] \cr & {I_{avg}} = 60\left[ {\frac{1}{{30\pi }}} \right] + \frac{2}{\pi } \cr & {I_{avg}} = \frac{4}{\pi }{\text{Amperes}} \cr & \cr & \left( {\text{b}} \right){\text{ }}0 \leqslant t \leqslant \frac{1}{{240}} \cr & {I_{avg}} = \frac{1}{{1/240 - 0}}\left[ { - \frac{{\cos \left( {60\pi t} \right)}}{{30\pi }} + \frac{{\sin \left( {120\pi t} \right)}}{{120\pi }}} \right]_0^{1/240} \cr & {I_{avg}} = 240\left[ { - \frac{{\cos \left( {60\left( {1/240} \right)\pi } \right)}}{{30\pi }} + \frac{{\sin \left( {120\left( {1/240} \right)\pi } \right)}}{{120\pi }}} \right] \cr & {I_{avg}} = 60\left[ { - \frac{{\cos \left( {\pi /4} \right)}}{{30\pi }} + \frac{{\sin \left( {\pi /2} \right)}}{{120\pi }}} \right] - 60\left[ { - \frac{{\cos \left( 0 \right)}}{{30\pi }}} \right] \cr & {I_{avg}} = 60\left[ { - \frac{{\sqrt 2 }}{{60\pi }} + \frac{1}{{120\pi }}} \right] + \frac{2}{\pi } \cr & {I_{avg}} = - \frac{{\sqrt 2 }}{\pi } + \frac{1}{{2\pi }} + \frac{2}{\pi } \cr & {I_{avg}} = \frac{{ - 2\sqrt 2 + 1 + 4}}{{2\pi }} \cr & {I_{avg}} = \frac{{5 - 2\sqrt 2 }}{{2\pi }}{\text{Amperes}} \cr & \cr & \left( {\text{c}} \right){\text{ }}0 \leqslant t \leqslant \frac{1}{{30}} \cr & {I_{avg}} = \frac{1}{{1/30 - 0}}\left[ { - \frac{{\cos \left( {60\pi t} \right)}}{{30\pi }} + \frac{{\sin \left( {120\pi t} \right)}}{{120\pi }}} \right]_0^{1/30} \cr & {I_{avg}} = 30\left[ { - \frac{{\cos \left( {60\left( {1/30} \right)\pi } \right)}}{{30\pi }} + \frac{{\sin \left( {120\left( {1/30} \right)\pi } \right)}}{{120\pi }}} \right] \cr & {I_{avg}} = 30\left[ { - \frac{{\cos \left( {2\pi } \right)}}{{30\pi }} + \frac{{\sin \left( {4\pi } \right)}}{{120\pi }}} \right] - 30\left[ { - \frac{{\cos \left( 0 \right)}}{{30\pi }}} \right] \cr & {I_{avg}} = 60\left[ { - \frac{1}{{60\pi }} + \frac{0}{{120\pi }}} \right] + \frac{1}{\pi } \cr & {I_{avg}} = - \frac{1}{\pi } + \frac{1}{\pi } \cr & {I_{avg}} = 0{\text{ Amperes}} \cr} $$