Question

the velocity of an object moving in a straight line, in meters per minute, can be modeled by the function ( v(t) ), where ( t ) is measured in minutes. selected values of ( v(t) ) are shown in the table below. approximate the value of ( \frac{1}{12}int_{0}^{12} v(t)dt ) using a midpoint riemann sum with 3 subintervals of equal length. you may use a calculator if necessary.

( t )	0	2	4	6	8	10	12

Explanation:

Step1: Find subinterval width

The interval is $[0,12]$, split into 3 equal subintervals.
Width $\Delta t = \frac{12-0}{3}=4$

Step2: Identify subintervals & midpoints

Subintervals: $[0,4], [4,8], [8,12]$
Midpoints: $t_1=2$, $t_2=6$, $t_3=10$

Step3: Calculate midpoint Riemann sum

Sum = $\Delta t \cdot [v(2)+v(6)+v(10)]$
$=4 \cdot (8+3+10)=4 \cdot 21=84$

Step4: Compute the final expression

Find $\frac{1}{12}$ of the sum:
$\frac{1}{12} \times 84$

Answer:

$7$