1. The velocity is constant and the distance is:
3.1 Accumulated Changes
Example 1: An objects travels with a velocity of 15 mph. What is the distance traveled after 4 hours
The velocity is constant and the distance is:
S = v.t = (4 ) (15) = 60 miles v
Distance = area = (4)(15) =60 15 t 1 2 3 4

2. Distance = area = 15 + 20 + 80 = 115 miles
Example 2: An objects travels for 4 hours with the following velocities:
First hour : 15 mph, Second hour: 20 mph , Last 2 hours: 40 mph What is the distance traveled
First hour : (15 mph)(1 hour) = 15 miles
Second hour: (20 mph)(1 hour) = 20 miles
Total = 115 miles
Last 2 hours: (40 mph)(2 hours) = 80 miles v
A=80 40
A=20
Distance = area = = 115 miles 20
A=15 15 t 1 2 3 4

3. Example 3: An objects travels with the following velocity: v = 15t + 10
Find the distance traveled after 4 hours t 1 2 3 4
Try t = 0,1,2,3,4 hours v 10 25 40 55 70
Total From Left: v 70
(10)(1) +
(25)(1) +
(40)(1) +
(55)(1) =130 70 55 55 40 40 25 25 10 t 1 2 3 4
 = 1 hour

4. Total From Left from previous slide:
Example 3 (Cont.): Using the same data and graph:
Total From Left from previous slide:
(10)(1) +
(25)(1) +
(40)(1) +
(55)(1) =130 v 70 70
Total From Right: 55 55
(70)(1) +
(55)(1) +
(40)(1) +
(25)(1) =190 40 40 25
Average: ( )/2 =160 miles 25 10 t 1 2 3 4
 = 1 hour

5. n is the number of intervals
Note:
is the interval size
n is the number of intervals v 70 70 70 55 55 55 40 40 40 25 25 25 10 t t 1 2 3 4 1
1.5 2 3 4
= 1 hour
n = 4 intervals
= 0.5 hour
n = 8 intervals

6. Example 4: An objects travels with the following velocity: v = 2t2 + 5
Find the distance traveled between t = 0 and t = 6 using n = 4 intervals t
1.5 3
4.5 6
= 6 hours / 4 intervals
=1.5 hour v 5
9.5 23
45.5 77
Total From Left: v 77
(5)(1.5) +
(9.5)(1.5) +
45.5
(23)(1.5) + 23
(45.5)(1.5) +
9.5
=124.5 5 t
1.5 3
4.5 6
 = 1.5 hour

7. Total From Left from previous slide:
Example 4 (Cont.): Using the same data and graph:
Total From Left from previous slide:
=124.5
Total From Right: v 77
(77)(1.5) +
(45.5)(1.5) +
45.5
(23)(1.5) + 23
(9.5)(1.5) +
9.5
=232.5 5 t
1.5 3
4.5 6
Average: ( )/2 =178.5 miles
 = 1.5 hour