Answer
The object makes 10.3 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.
$\theta = 65~rad$
We can convert the angle $\theta$ to revolutions.
$\theta = (65~rad)(\frac{1~rev}{2\pi~rad})$
$\theta = 10.3~rev$
The object makes 10.3 revolutions during the first 4 seconds.
