Chapter 8.1, Problem 28E

To determine

Thee connection between trapezoidal rule and the definite integral.

Answer to Problem 28E

The connection between this process and the definite integral is found by solving the integral

  ${\displaystyle \underset{0}{\overset{x}{\int }}\left|v\left(t\right)\right|dt}$ .

Explanation of Solution

Given information:

Anna accompanies her mother on a trip to the grocery store and keeps a log of the car’s speed at $10\sec$ intervals.

By measuring the speed of her mother’s car at $10\sec$ intervals,

For instances, Anna produces a table like the one listed below:

Time	Rate
$10$	$100$
$20$	$150$
$30$	$200$
$40$	$250$

Then by using Trapezoidal rule, she can estimate the total distance travelled by the car.

The connection between this process and the definite integrals is that Anna is trying to estimate the function $v\left(t\right)$ or the velocity of the car at a given time $t$ .

Hence, if the trip to the grocery shop took $x$ minutes, then the integral which gives Anna total distance travelled by the car can be found by solving the integral:

  ${\displaystyle \underset{0}{\overset{x}{\int }}\left|v\left(t\right)\right|dt}$ .