- SPEEDOMETER READINGS FOR A VEHICLE AT FOUR-SECOND INTERVALS
ARE GIVEN IN THE TABLE
WHERE WE HAVE THE TIME IN SECONDS
AND THE VELOCITY IN FEET PER SECONDS.
WE WANT TO ESTIMATE THE DISTANCE TRAVELED BY THE VEHICLE
DURING THE 20-SECOND PERIOD
USING THE VELOCITIES AT THE BEGINNING OF THE TIME INTERVALS
WHICH WILL BE THE LOWER SUM
BECAUSE THE VELOCITIES ARE INCREASING,
AND THEN WE'LL GIVE ANOTHER ESTIMATE
USING THE VELOCITIES AT THE END OF THE TIME INTERVALS,
AND AGAIN, BECAUSE THE VELOCITIES ARE INCREASING
THIS WOULD BE CALLED THE UPPER SUM.
SINCE DISTANCE = RATE x TIME,
WE CAN ESTIMATE THE DISTANCE TRAVELED
BY DETERMINING THE AREA OF RECTANGLES
OVER THESE SUB INTERVALS
USING THE VELOCITIES AS THE HEIGHT OF THE RECTANGLES.
SO TO BETTER UNDERSTAND THIS,
LET'S FIRST PLOT THESE POINTS ON THE COORDINATE PLANE.
HERE THEY ARE ON THE RIGHT.
FOR THE NEXT STEP,
LET'S GO AHEAD AND DIVIDE THIS 20-SECOND TIME INTERVAL
INTO 4-SECOND SUBINTERVALS.
AND NOW FOR THE FIRST ESTIMATE WE WANT TO USE THE VELOCITIES
AT THE BEGINNING OF EACH TIME INTERVAL
WHICH MEANS WE'D USE THE LEFT SIDE OF EACH SUBINTERVAL
TO DETERMINE THE HEIGHT OF THE RECTANGLE
THAT WE'LL USE TO APPROXIMATE THE DISTANCE.
SO LOOKING AT THIS FIRST SUBINTERVAL
WE'LL USE THE VELOCITY OF 24 OR THE LEFT SIDE HERE
AS THE HEIGHT OF THE FIRST RECTANGLE.
NOTICE HOW THE AREA OF THIS RECTANGLE WOULD BE 24 x 4
OR 24 FEET PER SECOND x 4 SECONDS,
GIVING US AN APPROXIMATION OF 96 FEET
FOR THIS FIRST 4-SECOND SUBINTERVAL.
CONTINUING THIS PATTERN,
WE'LL USE THE VELOCITY OF 30
FOR THE HEIGHT OF THE NEXT RECTANGLE.
NEXT, WE'LL USE THE VELOCITY OF 36
FOR THE HEIGHT OF THE NEXT RECTANGLE,
40 FOR THE NEXT HEIGHT,
AND 45 FOR THE LAST HEIGHT.
SO OUR APPROXIMATION FOR THE DISTANCE TRAVELED
WOULD BE THE TOTAL AREA OF THESE FIVE RECTANGLES.
I DO WANT TO POINT OUT THAT THE ACTUAL DISTANCE TRAVELED
WOULD BE THE AREA UNDER THE CURVE
THAT PASSES THROUGH THE 6 POINTS.
NOTICE HOW THE AREA UNDER THIS CURVE WOULD BE GREATER
THAN THE AREA OF THESE FIVE RECTANGLES,
AND THAT'S THE REASON WHY THIS IS CALLED THE LOWER SUM.
SO THE APPROXIMATION FOR THE DISTANCE TRAVELED
USING THE VELOCITIES AT THE BEGINNING OF EACH TIME INTERVAL
WOULD BE (24 x 4) + (30 x 4)
+ (36 x 4) + (40 x 4)
AND THEN FINALLY PLUS (45 x 4).
SO THIS WOULD GIVE US 96 + 120 + 144 + 160 + 180
WHICH IS EQUAL TO 700.
SO USING THIS APPROXIMATION
THE APPROXIMATE DISTANCE TRAVELED IS 700 FEET.
NOW WE'RE ASKED TO MAKE A SECOND APPROXIMATION NOW
USING THE VELOCITIES AT THE END OF THE TIME INTERVALS.
WE'LL FIRST FORM THE SUBINTERVALS,
AND NOW BECAUSE WE'RE USING THE VELOCITIES
FROM THE END OF EACH SUBINTERVAL
WE'RE USING THE RIGHT SIDE
TO FORM THE HEIGHT OF THE RECTANGLES.
SO FOR THIS FIRST SUBINTERVAL
WE'RE USING THE VELOCITY OF 30 AS THE HEIGHT OF THE RECTANGLE.
NEXT WE'RE USING THE VELOCITY OF 36 AS THE HEIGHT.
NEXT WE'LL USE THE VELOCITY OF 40 AS THE HEIGHT.
NEXT THE VELOCITY OF 45
AND FINALLY A VELOCITY OF 48 AS THE HEIGHT.
SO NOW WE'LL FIND THE AREA OF THESE FIVE RECTANGLES
TO MAKE ANOTHER APPROXIMATION FOR THE DISTANCE TRAVELED.
NOW THE REASON THIS WOULD BE CALLED THE UPPER SUM
IS BECAUSE IF WE SKETCHED A CURVE
THAT PASSED THROUGH THESE SIX POINTS,
THESE POINTS HERE, NOTICE HOW THE AREA UNDER THE CURVE,
WHICH WOULD BE THE EXACT DISTANCE,
WOULD BE LESS THAN THE AREA OF THE FIVE RECTANGLES.
SO THE APPROXIMATION USING THE ENDING VELOCITIES
WOULD BE APPROXIMATELY (30 x 4) + (36 x 4) + (40 x 4) + (45 x 4)
AND THEN FINALLY + (48 x 4).
SO WE'D HAVE 120 + 144 + 160 + 180 AND + 192
WHICH GIVES A SUM OF 796.
SO USING THESE VELOCITIES AS THE HEIGHT OF THE RECTANGLES
THE APPROXIMATE DISTANCE TRAVELED IS 796 FEET.
SO IN REALITY THE DISTANCE TRAVELED
WOULD MORE THAN LIKELY BE SOMEWHERE BETWEEN 796 FEET
AND OUR PREVIOUS ESTIMATE OF 700 FEET.
I HOPE YOU FOUND THIS HELPFUL.