MATH 152 Chapter 7.5

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Average Value

...of a finite number of $n$ values:

${\frac {1}{n}}\sum _{i=1}^{n}{x_{i}}$

...of a function with a finite number of equally-spaced partitions between $a$ and $b$:

$f_{\text{avg}}={\frac {1}{n}}\sum _{i=1}^{n}{f(x_{i}^{*})\,\Delta x_{i}^{*}}={\frac {1}{b-a}}\int _{a}^{b}{f(x)\,\mathrm {d} x}$

If $f$ > 0 from $a$ to $b$, then $(b-a)f_{\text{avg}}=\int _{a}^{b}{f(x)\,\mathrm {d} x}$

Warm-up

The electric current in a household power supply is an alternating current modeled by $i(t)=I\,\sin {\omega t}$

a) Show that the average value of $i$ over one period is zero

${\text{one period}}=[0,{\frac {2\pi }{\omega }}]$
$i_{\text{avg}}={\frac {1}{{\frac {2\pi }{\omega }}-0}}\,\int _{0}^{\frac {2\pi }{\omega }}{I\,\sin {\omega t}\,\mathrm {d} t}=0$

b) The root mean square (rms) current is the square root of the average value of $i^{2}$ over one period. Calculate the rms current

$i_{\text{rms}}={\sqrt {{\frac {1}{{\frac {2\pi }{\omega }}-0}}\,\int _{0}^{\frac {2\pi }{\omega }}{(I\,\sin {\omega t})^{2}\,\mathrm {d} t}}}$

$i_{\text{rms}}={\sqrt {{\frac {\omega I^{2}}{4\pi }}\,\int _{0}^{\frac {2\pi }{\omega }}{1-\cos {2\omega t}\,\mathrm {d} t}}}={\frac {I}{\sqrt {2}}}$