Your task is to estimate how far an object traveled during the time interval 0 ≤ t ≤ 8, but you only have the following data about the velocity of the object.

Question

Your task is to estimate how far an object traveled during the time interval 0 ≤ t ≤ 8, but you only have the following data about the velocity of the object.

time (sec)	0	1	2	3	4	5	6	7	8
velocity (feet/sec)	-4	-3	-1	-4	-3	1	4	1	3

To get an idea of what the velocity function might look like, you pick up a black pen, plot the data points, and connect them by curves. Your sketch looks something like the black curve in the graph below.

Left endpoint approximation

You decide to use a left endpoint Riemann sum to estimate the total displacement. So, you pick up a blue pen and draw rectangles whose height is determined by the velocity measurement at the left endpoint of each one-second interval. By using the left endpoint Riemann sum as an approximation, you are assuming that the actual velocity is approximately constant on each one-second interval (or, equivalently, that the actual acceleration is approximately zero on each one-second interval), and that the velocity and acceleration have discontinuous jumps every second. This assumption is probably incorrect because it is likely that the velocity and acceleration change continuously over time. However, you decide to use this approximation anyway since it seems like a reasonable approximation to the actual velocity given the limited amount of data.

(A) Using the left endpoint Riemann sum, find approximately how far the object traveled. Your answers must include the correct units.

Total displacement =

Total distance traveled =

Using the same data, you also decide to estimate how far the object traveled using the right endpoint Riemann sum. So, you sketch the curve again with a black pen and draw rectangles whose height is determined by the velocity measurement at the right endpoint of each one-second interval.

Right endpoint approximation

(B) Using the right endpoint Riemann sum, find approximately how far the object traveled. Your answers must include the correct units.

Total displacement =

Total distance traveled =

Answer 1

The calculations below are not left point or rjght point sums but very close to mid point if the velocity is a linear function of time between seconds. The interval 4s. to 5s. creates a special problem. Notice acceleration is changing at integer t moments abruptly (dv/dt), but velocity is continuous at those points. I drew lines creating triangles plus boxes for area calculation. Then for sure acceleration is discontinuous at those points.

If the idea is only to use boxes for area for the left and right sums, then adding up shaded boxes

might be the simple way of answering without estimating any closer.

the right point box areas add to -2. the left point box areas add to -9 in movement sum.

For displacement, right box add without regard to direction to 20. Displacement left box method =21.

I calculated movement equaled -5.75 in sum. Without regard to direction displacement =20.5

remember to include both box area and triangle areas for each second.

For time 4s. to 5s. it is a bit tricky. There is negative area from 4s. to 4.75s.

and positive area from 4.75s. to 5s.. The triangle of both with base 4s. to 5s.

is 4 high so has area 2 (area is movement in units and in calculus). Positive area

is (1/2)*.25*1=.125 so negative area is 2-.125

To find where x is when the triangle hypotenuse crosses x-axis, must

use slope=4=(y+3)/(x-4),,,and set y=0