Integrating

Given the fact that we can differentiate to get velocity and accelerations, it follows that we can integrate to "go backwards".

Example

At time $t = 0$, an object sets off from the rest at the origin, $O$, and travels along the x-axis. At the time $t$ seconds, the object is accelerating at the rate of $(2t - 4) \text{ ms}^{-2}$ Find the time it takes to return to the point $O$.

Integrate to find the displacement

$$\begin{align} a &= 2t - 4\\ v &= \int{a \text{ }dt} = t^2 - 4t\\ s &= \int{v \text{ }dt} = \frac{1}{3}t^3 - 2t^2 \end{align}$$ Not we can eliminate the constants of integration since we know the acceleration is zero when the time is zero and that the displacement is zero when time is zero.

Find the point where the displacement is zero

$$\begin{align} \frac{1}{3}t^3 - 2t^2 &= 0\\ t^2 \left(\frac{1}{3}t - 2\right) &= 0\\ t &= 0, 6\\ t &= 6 \end{align}$$