Given the velocity function v(t) = 3t^2 - 6t, calculate the displacement of the object from t = 1 to t = 4.

Given the acceleration function a(t) = 6t - 4 and initial velocity v(0) = 3, find the velocity function v(t).

A particle moves along a line with velocity $v\left(t\right)=4\sin \left(2\pi t\right)$, and initial position $s\left(0\right)=0$. Find the displacement of the particle from $t=0$ to $t=1$, and the total distance traveled during this interval.

Two cars, $M$ and $N$, start from the same point with velocities $v_M\left(t\right)=2\sin t$ and $v_N\left(t\right)=2\cos t$, respectively, both with initial positions of zero. At what time in $[0,2\pi ]$ do they first meet, and what is their position at that time?

A fish population in a lake is initially $F\left(0\right)=120$. The growth rate is given by $F^{\prime }\left(t\right)=25-\frac{t}{8}$ (fish/month), for $0\le t\le 160$, where $t$ is in months. What is $F\left(40\right)$?