Answer
A lower estimate for the distance that the runner traveled is $34.7~feet$ during these three seconds.
An upper estimate for the distance that the runner traveled is $44.8~feet$ during these three seconds.
Work Step by Step
The interval $[0,3.0]$ is divided into 6 subintervals.
$\Delta t = \frac{b-a}{n} = \frac{3.0-0}{6} = 0.5$
To find a lower estimate for the distance, we can use the left endpoint of each subinterval:
$t_1 = 0$
$t_2 = 0.5$
$t_3 = 1.0$
$t_4 = 1.5$
$t_5 = 2.0$
$t_6 = 2.5$
We can find a lower estimate for the distance:
$\sum_{i=1}^{6} v(t_i)~\Delta t$
$= (0+6.2+10.8+14.9+18.1+19.4)~(0.5)$
$= 34.7$
A lower estimate for the distance that the runner traveled is $34.7~feet$ during these three seconds.
To find an upper estimate for the distance, we can use the right endpoint of each subinterval:
$t_1 = 0.5$
$t_2 = 1.0$
$t_3 = 1.5$
$t_4 = 2.0$
$t_5 = 2.5$
$t_6 = 3.0$
We can find an upper estimate for the distance:
$\sum_{i=1}^{6} v(t_i)~\Delta t$
$= (6.2+10.8+14.9+18.1+19.4+20.2)~(0.5)$
$= 44.8$
An upper estimate for the distance that the runner traveled is $44.8~feet$ during these three seconds.