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- IN THIS LESSON WE'LL TAKE A LOOK
AT SEVERAL EXAMPLES OF INTERPRETING THE MEANING
OF THE AREA UNDER A CURVE.
SO FOR THIS EXAMPLE WE WANT TO INTERPRET THE AREA
OF THIS BLUE SHADED REGION.
THE KEY TO DOING THESE TYPES OF PROBLEMS IS IDENTIFYING
THE UNITS ON THE X AND Y AXIS.
NOTICE THE X AXIS REPRESENTS THE TIME AND HOURS,
AND THE Y AXIS REPRESENTS THE NUMBER OF KILOWATTS
USED PER HOUR.
AND IT MAY BE HELPFUL TO REPRESENT A SLICE
OF THIS AREA WITH A RECTANGLE, MAYBE SOMETHING LIKE THIS.
AND THEN TO DETERMINE THE AREA OF THIS RECTANGLE,
WE HAVE TO MULTIPLY THE WIDTH x THE HEIGHT
WHERE THE WIDTH REPRESENTS THE TIME AND HOURS
AND THE HEIGHT REPRESENTS THE KILOWATTS USED PER HOUR.
SO TO DO THESE TYPES OF PROBLEMS WE NEED TO INTERPRET
WHAT IT MEANS WHEN WE MULTIPLY TIME AND HOURS x KILOWATTS
USED PER HOUR.
SO IN THIS CASE THE ENTIRE SHADED REGION
REPRESENTS THE NUMBER OF KILOWATTS USED
OVER THIS TIME PERIOD OF EIGHT HOURS.
NOTICE HOW THE USAGE PER HOUR DOES VARY.
IT STARTS LOW, INCREASES, AND THEN DECREASES,
BUT THE AREA REPRESENTS THE TOTAL KILOWATTS
USED OVER THIS TIME PERIOD.
FOR THE SECOND EXAMPLE, NOTICE THAT THE X AXIS
REPRESENTS TIME AND HOURS AGAIN, BUT NOW THE Y AXIS REPRESENTS
THE VELOCITY IN MILES PER HOUR.
SO, AGAIN, IF IT'S HELPFUL, WE CAN TAKE A SLICE
OF THIS AREA, AND THEN TO DETERMINE THE AREA
OF THIS RECTANGLE, WE HAVE TO MULTIPLY
THE WIDTH OR THE TIME AND HOURS x THE HEIGHT,
WHICH WOULD BE THE VELOCITY IN MILES PER HOUR.
SO WE NEED TO INTERPRET THE MEANING BEHIND MULTIPLYING
THE TIME AND HOURS BY THE VELOCITY
AND MILES PER HOUR.
WELL, VELOCITY IS A RATE.
WHEN WE MULTIPLY A RATE AND TIME WE ACTUALLY GET DISTANCE.
SO THIS ORANGE SHADED REGION REPRESENTS THE TOTAL DISTANCE
TRAVELED OVER FOUR HOURS.
THIS SHOULD REMIND US OF THE FORMULA
DISTANCE = RATE x TIME WHERE THE Y AXIS IS THE VELOCITY
OR THE RATE, AND THE X AXIS IS THE TIME AND HOURS.
LET'S TAKE A LOOK AT ONE MORE EXAMPLE.
THE X AXIS REPRESENTS THE NUMBER OF DAYS
AND THE Y AXIS REPRESENTS THE REVENUE PER DAY.
SO WE COULD SKETCH A RECTANGLE THAT REPRESENTS
A PIECE OF THIS AREA TO HELP US INTERPRET THE MEANING
OF THE TOTAL AREA.
TO DETERMINE THE AREA OF THIS RECTANGLE WE'D MULTIPLY
THE WIDTH, WHICH REPRESENTS THE NUMBER OF DAYS,
x THE HEIGHT, WHICH REPRESENTS THE REVENUE PER DAY.
WELL, IF WE MULTIPLY THE NUMBER OF DAYS x THE REVENUE PER DAY,
THE RESULT WOULD BE THE TOTAL REVENUE OVER THAT TIME PERIOD.
SO THIS GREEN AREA REPRESENTS THE TOTAL REVENUE GENERATED
OVER 20 DAYS.
SO THE KEY TO DOING THESE TYPES OF PROBLEMS
IS INTERPRETING THE MEANING BEHIND MULTIPLYING THE UNITS
ON THE X AXIS BY THE UNITS ON THE Y AXIS.
I HOPE THIS WAS HELPFUL.