Angular Displacement of bike pedals

[snowmx0090]

An exhausted bicyclist pedals somewhat erratically, so that the angular velocity of his tires follows the equation,
ω(t)=(1/2)t - (1/4)sin(2t)
where represents time (measured in seconds).
There is a spot of paint on the front tire of the bicycle. Take the position of the spot at time to be at angle radians with respect to an axis parallel to the ground (and perpendicular to the axis of rotation of the tire) and measure positive angles in the direction of the tire's rotation. What angular displacement has the spot of paint undergone between time 0 and 2 seconds?
Express your answer in radians.

I thought you would take the integral of the equation from t=0 to t=2 in which the equation would then be (1/4)t^2 + (1/2)cos(2t) but this was wrong. Where should I start with this equation?

 

[Doc Al]

I thought you would take the integral of the equation from t=0 to t=2 in which the equation would then be (1/4)t^2 + (1/2)cos(2t) but this was wrong.

Check your integration--the second term has an error.

 

[snowmx0090]

Would the equation after integrating be (1/4)t^2 + (1/4)cos(2t)? Was this my mistake?

 

[radou]

Would the equation after integrating be (1/4)t^2 + (1/4)cos(2t)? Was this my mistake?

There is still a mistake. What does $$\frac{d}{dt}(\frac{1}{4}\cos (2t))$$ equal?

 

[faller217]

you would have to use a u substitute. Set u=2t, and du=2. So instead of it being 1/4cos(2t), it would be 1/8cos(2t).

final : ((1/4)(t^2))+((1/8)(cos2t)) from 0 to 2 seconds

.918295 - .125 = .793 radians