Starting from rest, a wheel is uniformly accelerated at 4 rad/s^2...

The motion of the wheel can be divided into three phases.

In the first phase, the wheel starts from rest and accelerates uniformly.
Initial angular velocity, $\omega_0 = 0$
Angular acceleration, $\alpha_1 = 4 \text{ rad/s}^2$
Time, $t_1 = 10 \text{ s}$
Angle rotated, $\theta_1 = \omega_0 t_1 + \dfrac{1}{2} \alpha_1 t_1^2 = 0 + \dfrac{1}{2} \times 4 \times (10)^2 = 200 \text{ rad}$
Angular velocity at the end of this phase, $\omega_1 = \omega_0 + \alpha_1 t_1 = 0 + 4 \times 10 = 40 \text{ rad/s}$

In the second phase, the wheel rotates uniformly.
Angular velocity, $\omega_1 = 40 \text{ rad/s}$
Time, $t_2 = 10 \text{ s}$
Angle rotated, $\theta_2 = \omega_1 t_2 = 40 \times 10 = 400 \text{ rad}$

In the third phase, the wheel is brought to rest uniformly.
Initial angular velocity, $\omega_1 = 40 \text{ rad/s}$
Final angular velocity, $\omega_2 = 0$
Time, $t_3 = 10 \text{ s}$
Angle rotated, $\theta_3 = \left( \dfrac{\omega_1 + \omega_2}{2} \right) t_3 = \left( \dfrac{40 + 0}{2} \right) \times 10 = 200 \text{ rad}$

Total angle rotated by the wheel is $\theta = \theta_1 + \theta_2 + \theta_3 = 200 + 400 + 200 = 800 \text{ rad}$.