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- WE WANT TO FIND THE VALUE OF THE RIEMANN SUM GIVEN HERE
FOR F OF X = 2 RAISED TO THE POWER OF X + 10
USING THE PARTITIONS P = 0, 2, 5, AND 7 WHERE C SUB K,
THE INPUT INTO THE FUNCTION IS THE LEFT ENDPOINT
OF EACH PARTITION.
SO BECAUSE OUR FUNCTION IS NON-NEGATIVE
OVER THIS INTERVAL WE CAN THINK OF THIS RIEMANN SUM
AS SUMMING THE AREA OF RECTANGLES
TO APPROXIMATE THE AREA UNDER THE CURVE WHERE F OF C x DELTA X
WOULD GIVE US THE AREA OF A RECTANGLE
WHERE F OF C WOULD BE THE HEIGHT OF THE RECTANGLE
AND DELTA X WOULD BE THE WIDTH OF THE RECTANGLE.
IF THE FUNCTION WAS NEGATIVE OR IT DROPPED BELOW THE X-AXIS
WE COULD USE THE SAME PROCESS,
BUT IT WOULD NOT REPRESENT THE AREA
BECAUSE IF THE FUNCTION IS BELOW THE X-AXIS
THE FUNCTION VALUE FOR F OF C WOULD BE NEGATIVE.
LET'S BEGIN BY SETTING UP OUR PARTITIONS USING 0, 2, 5, AND 7.
SO HERE'S X = 0, HERE'S X = 2, HERE'S X = 5, AND HERE'S X = 7.
NOTICE HOW USING THESE 4 VALUES WE HAVE THREE PARTITIONS.
THIS FIRST PARTITION HAS A WIDTH OF 2 WHICH WOULD BE DELTA X.
SECOND PARTITION HAS A WIDTH OF 3 WHICH WOULD BE DELTA X,
AND THE THIRD PARTITION HAS A WIDTH OF 2
WHICH AGAIN WOULD BE DELTA X.
NOW WE'LL SKETCH OUR RECTANGLES
USING THE LEFT ENDPOINT OF EACH PARTITION.
SO FOR THIS FIRST PARTITION F OF 0 WOULD BE THE HEIGHT
AND THEREFORE THIS WOULD BE OUR RECTANGLE.
FOR THE SECOND PARTITION, AGAIN USING THE LEFT SIDE,
THIS WOULD BE THE HEIGHT OF THE RECTANGLE.
AND FOR THIS THIRD PARTITION, AGAIN USING THE LEFT SIDE,
THIS WOULD BE THE HEIGHT OF THE RECTANGLE.
NOTICE HOW THE AREA OF THE RECTANGLES
WOULD BE LESS THAN THE AREA UNDER THE CURVE
AND ABOVE THE X-AXIS,
BUT THE VALUE OF THIS RIEMANN SUM
WOULD BE THE SUM OF THESE THREE AREAS
OR THE AREA OF THIS SHADED REGION.
LET'S GO AHEAD AND SET THIS UP USING FUNCTION NOTATION FIRST.
WE HAVE V = AGAIN, F OF C x DELTA X.
WE'RE AGAIN USING THE LEFT ENDPOINT.
WE'D HAVE F OF 0 x DELTA X WHICH IS 2 UNITS
PLUS FOR THE NEXT RECTANGLE THE HEIGHT IS AGAIN THE LEFT SIDE
OR F OF 2 x DELTA X WHICH IS 3
PLUS THE HEIGHT OF THE LAST RECTANGLE
WOULD BE F OF 5 x DELTA X, THE WIDTH, WHICH IS 2 UNITS.
FOR THE NEXT STEP WE'LL DETERMINE THE FUNCTION VALUES,
SO F OF 0 WOULD BE 2 TO THE 0.
THAT'S GOING TO BE 1 + 10, THAT'S 11.
SO WE HAVE 11 x 2 + F OF 2 WOULD BE 2 SQUARED.
THAT'S 4 + 10.
THAT'S GOING TO BE 14, SO 14 x 3 + F OF 5 WOULD BE 2 TO THE 5TH,
THAT'S 32 + 10, THAT'S 42 x 2.
SO WE HAVE 22 + 42 + 84.
THE VALUE OF THIS RIEMANN SUM IS 148.
AND IF WE WERE USING THIS TO APPROXIMATE THE AREA
UNDER THE CURVE,
AGAIN, THIS WOULD NOT BE A VERY GOOD APPROXIMATION
BECAUSE WE CAN SEE THERE'S A LOT OF AREA THAT'S LEFT OUT
THAT'S UNDER THE CURVE
BUT NOT PART OF THE AREA OF THE RECTANGLES.
I HOPE YOU FOUND THIS HELPFUL.