Question

explain in words what the integral represents and give units.
\\( \int _ { 2 } ^ { 9 } v ( t ) d t \\),
where \\( v ( t ) \\) is velocity in meters/sec and \\( t \\) is time in seconds.
the integral represents the
the units are

Explanation:

The definite integral of velocity \(v(t)\) over an interval of time \([a,b]\) (in this case \(a = 2\) and \(b=9\)) represents the net - change in position (displacement). This is because, by the fundamental theorem of calculus, if \(s(t)\) is the position function and \(s^\prime(t)=v(t)\), then \(\int_{a}^{b}v(t)dt=s(b)-s(a)\).

For the units, when we multiply velocity (meters/sec) by time (sec), using the formula for the integral \(\int_{2}^{9}v(t)dt\approx\sum_{i = 1}^{n}v(t_i)\Delta t\) (Riemann sum approximation), the units of \(v(t)\) is meters/sec and the units of \(dt\) (or \(\Delta t\)) is sec. So, \((\text{meters/sec})\times(\text{sec})=\text{meters}\).

Answer:

The integral represents the displacement (change in position) of an object from \(t = 2\) seconds to \(t=9\) seconds. The units are meters.