The instantaneous velocity $v$ is defined as the rate of change of displacement $s$: $v=\frac{ds}{dt}⟹ds=vdt$ Integrating both sides over a time interval from $t_1$ to $t_2$: $\int ds={\int }_{t_1}^{t_2}vdt$ $s=\text{Area under the }v-t\text{ graph}$. Note: If the velocity becomes negative, the 'area' below the axis is subtracted to give displacement. Total area (ignoring sign) gives distance.