Step 1 :A particle starts out from the origin. Its velocity, in miles per hour, after t hours is given by \(v(t)=15 t^{2}+2 t\). We are asked to find how far it travels from the 2nd hour through the 6th hour.
Step 2 :The distance traveled by the particle is the integral of the velocity function over the given time interval. In this case, we need to integrate the function \(v(t)=15 t^{2}+2 t\) from \(t=2\) to \(t=6\).
Step 3 :The result of the integration gives the total distance traveled by the particle from the 2nd hour through the 6th hour.
Step 4 :Final Answer: The particle travels \(\boxed{1072}\) miles from the 2nd hour through the 6th hour.