The following table gives the velocity v(t) (in feet per sec) at different instances of time t (in sec) of a particle moving along a horizontal axis.

t 0 4 8 12 16 20
v(t) 43 42 40 35 25 5

Estimate the distance traveled by the particle between t = 0 sec and t = 20 sec using 5 time intervals of equal length with

(a) left endpoints
Estimated distance = _____feet

(b) right endpoints
Estimated distance = _____feet

t 0 4 8 12 16 20

vt 43 42 40 35 25 5

left endpoints: 4*43 + 4*42 + ... + 4*25

right ends: 4*42 + 4*40 + ... + 4*5

hey thanks but i still don't get how you arrived to this answer?

consult your text, or do a web search. To approximate the area under a curve, draw rectangles of (in this case) width 4.

The vertical sides intersect the curve at various points. To approximate using left endpoints, make the height of the rectangle spanning x0-x1 the value of f(x0) - the left height.

Thus, the area of rectangle from xi to xi+1 is w*f(xi)

Using the right endpoints of the intervals, the area is w*f(xi+1)

Takes a lot of words, but any illustration should make it clear.

Why are we approximating the area under the curve? Because we are given the speed at various times. speed is the derivative of the position. The position is thus the integral of the speed over an interval.

To estimate the distance traveled by the particle between t = 0 sec and t = 20 sec using equal time intervals with left or right endpoints, we can use the concept of Riemann sums. A Riemann sum is an approximation of the area under a curve by dividing the area into smaller rectangles. In this case, we will approximate the distance traveled by dividing the time interval into smaller subintervals and approximating the area between the velocity curve and the x-axis.

(a) To estimate the distance using left endpoints:
1. Divide the time interval [0, 20] into 5 equal subintervals. Since the time interval is 20 sec, each subinterval will have a width of 20/5 = 4 sec.
2. Set up a table to represent the subintervals and their corresponding velocities:

Subinterval Velocity (v)
[0, 4] 43
[4, 8] 42
[8, 12] 40
[12, 16] 35
[16, 20] 25

3. Multiply each velocity by the width of its corresponding subinterval to get the distance traveled in that subinterval:

[0, 4]: 43 * 4 = 172 ft
[4, 8]: 42 * 4 = 168 ft
[8, 12]: 40 * 4 = 160 ft
[12, 16]: 35 * 4 = 140 ft
[16, 20]: 25 * 4 = 100 ft

4. Sum up the distances traveled in each subinterval to estimate the total distance:

Σ Distance = 172 + 168 + 160 + 140 + 100 = 740 ft

Therefore, the estimated distance traveled by the particle between t = 0 sec and t = 20 sec using equal time intervals with left endpoints is 740 ft.

(b) To estimate the distance using right endpoints:
1. Follow steps 1 and 2 from part (a) to set up the table with the subintervals and corresponding velocities.
2. Find the velocity at the right endpoint of each subinterval instead of the left endpoint:

[0, 4]: 42 * 4 = 168 ft
[4, 8]: 40 * 4 = 160 ft
[8, 12]: 35 * 4 = 140 ft
[12, 16]: 25 * 4 = 100 ft
[16, 20]: 5 * 4 = 20 ft

3. Sum up the distances traveled in each subinterval to estimate the total distance:

Σ Distance = 168 + 160 + 140 + 100 + 20 = 588 ft

Therefore, the estimated distance traveled by the particle between t = 0 sec and t = 20 sec using equal time intervals with right endpoints is 588 ft.