Answer
The particle makes 9.5 revolutions during the first 4 seconds.
Work Step by Step
The area under the $\omega$ versus time graph is equal to $\theta$, the total angle through which the particle moves:
$\theta = (20~rad/s)(2~s)+(10~rad/s)(2~s)$
$\theta = 60~rad$
We can convert the angle $\theta$ to revolutions;
$\theta = (60~rad)(\frac{1~rev}{2\pi~rad})$
$\theta = 9.5~rev$
The particle makes 9.5 revolutions during the first 4 seconds.
