suppose the odometer on our car is broken and we want to estimate the distance driven over a 30 second time interval We take the speedometer readings every five seconds and record them in the following table Time s 0 5 10 15 20 25 30 Velocity mi h 16 20 22 29 31 30 28 In order to have the time and the velocity in consistent units let s convert the velocity readings to feet per second 1 mi h ft s Round your answers to the nearest whole number 5280 3600 5 Time s 0 Velocity ft s 23 10 15 32 During the first five seconds the velocity doesn t change very much so we can estimate the distance traveled during that time by assuming that the velocity is constant If we take the velocity during that time interval to be the initial velocity 25 ft s then we obtain the approximate distance traveled during the first five seconds If we had wanted ar 20 25 30 23 ft s x 5 s Similarly during the second time interval the velocity is approximately constant and we take it to be the velocity when t 5 s So our estimate for the distance traveled from t 5 s to t 10 s is 29 ft s x 5 s If we add similar estimates for the other time intervals we obtain an estimate for the total distance traveled ft 45 44 41 ft 23 x 5 29 x 5 32 x 5 43 x 5 45 x 5 44 x 5 We could just as well have used the velocity at the end of each time period instead of the velocity at the beginning as our assumed constant velocity Then our estimate becomes 29 5 32 5 43 5 45 5 44 x 5 41 x 5 ft ft e we could have taken velocity readings every two seconds or