Question

the angular velocity of a rotating wheel as a function of time

Answer 1

There are a few ways to do this. 

i) observe that the acceleration is 9rad/s / 3s = 3 rad/s², so 
Θ = ω₀*t + ½αt² = -9rad/s*10s + ½ * 3rad/s² * (10s)² = 60 rads 

ii) It gained 15 rad/s over 5 s, so by 10 s it will gain another 15 rad/s with a final velocity of 21 rad/s. Then 
Θ = ωavg * t = ½(21 - 9)rad/s * 10s = 60 rads 

iii) the area under the t-axis is ½ * -9rad/s * 3s = -13.5 rads 
and the area over is ½ * 21rad/s * 7s = 73.5 rads 
so the net area = angular distance = 60 rads