Define root mean square value of an alternating current.

If an analog sinusoidal voltage in a circuit as a function of time is
$\mathrm{v}={\mathrm{v}}_0\sin \omega \mathrm{t}\text{ for }0\le \mathrm{t}\le \pi /\omega$

$\mathrm{v}=-{\mathrm{v}}_0\sin \omega \mathrm{t}\text{ for }\pi /\omega \le \mathrm{t}\le 2\pi /\omega$
Then the average value of the voltage in complete cycle is (where symbols have their usual meanings)

Find the average for the saw-tooth voltage of peak value $V_0$ from $t=0$ to $t=2T$ as shown in the figure. 

The average value of voltage for a triangular wave having peak $V_0$, as shown in the figure, for time interval $t=0$ to $t=\frac{T}{2}$, is 

An inductor-coil having some resistance is connected to an AC source. Which of the following quantities have zero average value over a cycle? 

Represent the union of two sets by Venn diagram for each of the following.

$X=\{x | x$ is a prime number between $80$ and $100\}$

$Y=\{y | y$ is an odd number between $90$ and $100\}$