Newton's Second Law for Rotation

1. The problem statement, all variables and given/known data
A pulley, with a rotational inertia of 1.0 x 10^-3 kg*m^2 about its axle and a radius of 10 cm, is acted on tangentially at its rim. the force magnitude varies in time as F=0.50t + 0.30t^2, with F in newtons and t in seconds. The pulley is initially at rest . At t=3.0 s what are its (a) angular acceleration and (b) its angular speed?


2. Relevant equations

(a) [tex]\alpha[/tex]= (r*F(3)) / I = 420 rad/sec^2
(b) unsure, but tried using [tex]\omega[/tex]= [tex]\omega[/tex][tex]_{}0[/tex] +[tex]\alpha[/tex]t = 1260 rad/s which is wrong and several other variations of that equation.

3. The attempt at a solution

(see above)

 

so is the geometric relationship that when alpha increases, so does omega? do i need to use an integral?
dw= (integral)alpha dt? but how can i differentiate this if there are no variables?
i come up with dw=alpha*t + c but when i plug in t=3, i still get 1260...

 

they are just asking for alpha at t=3 s, which is 420 rad/s^2. I am fairly sure of this because it is the answer in the back of the book... However, i am struggling to figure out omega at t=3 s which should be 500 rad/s but i need the work to back it up and i'm not sure which equations to use.