What does the area under acceleration-time graph represent for any given time interval

The correct option is Change in the velocity in that time interval.

To understand why the area under an acceleration-time ($a-t$) graph represents the change in velocity, we can look at the definition of acceleration.

By definition, acceleration is the rate of change of velocity with respect to time:
$a=\frac{dv}{dt}$
where $a$ is the acceleration, $dv$ is an infinitesimal change in velocity, and $dt$ is an infinitesimal time interval.

We can rearrange this equation to express the small change in velocity:
$dv=a\cdot dt$

To find the total change in velocity over a finite time interval from $t_1$ to $t_2$, we integrate both sides of the equation:
${\int }_{v_1}^{v_2}dv={\int }_{t_1}^{t_2}a\cdot dt$

The left side of the equation simplifies to the change in velocity:
$\Delta v=v_2-v_1={\int }_{t_1}^{t_2}a\cdot dt$

In calculus, the definite integral of a function $a\left(t\right)$ from $t_1$ to $t_2$ corresponds geometrically to the area under the curve of $a\left(t\right)$ plotted against $t$ within that time interval.

Therefore, the area under the acceleration-time graph for a given time interval is equal to the integral of acceleration over that time, which physically represents the change in the velocity in that time interval.