A particle starts out from the origin. Its velocity, in miles per hour, after thours is given by $v(t)=15 t^{2}+2 t$. How far does it travel from the $2 n d$ hour through the 6 th hour $(t=1$ to $t=6)$ ?
From the 2nd hour through the 6 th hour it will travel $\mathrm{mi}$. (Simplify your answer.)

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.