- WE WANT TO FIND THE VALUE OF THE RIEMANN SUM GIVEN HERE
FOR F OF X = -X SQUARED + 16 USING THE PARTITIONS
P = 1, 3, 5, AND 8 WHERE C SUB K IS THE RIGHT ENDPOINT
OF EACH PARTITION.
LOOKING AT THE SUM FOR A MOMENT NOTICE HOW WE'LL BE SUMMING
THE PRODUCT OF F OF C AND DELTA X USING THESE PARTITIONS.
SO IF THE FUNCTION WAS NON- NEGATIVE OVER THESE PARTITIONS
WE COULD THINK OF THIS AS SUMMING THE AREA OF RECTANGLES
WHERE F OF C WOULD BE THE HEIGHT AND DELTA X WOULD BE THE WIDTH,
BUT BECAUSE OUR FUNCTION IS NEGATIVE WHEN X
IS GREATER THAN 4 WE WILL HAVE NEGATIVE VALUES FOR F OF C.
AND THEREFORE WE CAN'T REALLY THINK OF THIS AS AREA
UNLESS WE THINK OF THE AREA ABOVE THE X-AXIS AS POSITIVE
AND THE AREA BELOW THE X-AXIS AS NEGATIVE.
LET'S BEGIN BY SETTING UP THE PARTITIONS USING 1,3, 5, AND 8.
HERE'S WHERE X = 1, HERE'S WHERE X = 3, X = 5 HERE,
AND FINALLY X = 8.
NOTICE USING THESE VALUES HERE WE HAVE THREE PARTITIONS.
THE FIRST TWO HAVE A WIDTH OF 2 UNITS WHICH WOULD BE DELTA X.
THE THIRD PARTITION HAS A WIDTH OF 3 UNITS
WHICH WOULD BE DELTA X.
AND NOW WE'LL FORM RECTANGLES
USING THE RIGHT ENDPOINT OF EACH PARTITION.
LOOKING AT THIS FIRST PARTITION
THIS WOULD BE THE HEIGHT OF THE RECTANGLE,
THE RIGHT SIDE OF THE PARTITION.
BECAUSE THIS RECTANGLE IS ABOVE THE X-AXIS
WE CAN THINK OF F OF C x DELTA X AS THE AREA OF THIS RECTANGLE.
BUT FOR THE NEXT PARTITION USING THE RIGHT ENDPOINT,
NOTICE HOW THE FUNCTION VALUE HERE IS NEGATIVE,
SO WE'LL HAVE A RECTANGLE THAT'S BELOW THE X-AXIS
THAT LOOKS LIKE THIS.
AND BECAUSE THE FUNCTION VALUE F OF 5 WILL BE NEGATIVE
WE COULD THINK OF THIS AS A NEGATIVE AREA.
AND THEN FOR THE LAST PARTITION, AGAIN USING THE RIGHT ENDPOINT,
AGAIN THE RECTANGLE WOULD BE BELOW THE X-AXIS
WHERE WE CAN THINK OF THIS AS THE HEIGHT
EVEN THOUGH F OF 8 WOULD BE A NEGATIVE FUNCTION
VALUE AND THEREFORE F OF C x DELTA X IS GOING TO BE NEGATIVE.
SO WE CAN THINK OF THE AREA OF THIS RECTANGLE
AS A POSITIVE AREA,
AND IF IT'S HELPFUL WE CAN THINK OF THE AREA
OF THESE TWO RECTANGLES AS NEGATIVE
EVEN THOUGH WE KNOW AREA CAN NEVER BE NEGATIVE,
IT IS ONE WAY TO VIEW THIS.
ONE THING WE SHOULD NOTICE, THOUGH,
BECAUSE THE AREA OF THE RECTANGLES BELOW THE X-AXIS
IS GREATER THAN THE AREA OF THE RECTANGLE
ABOVE THE X-AXIS THE VALUE OF THIS RIEMANN SUM
SHOULD BE NEGATIVE.
LET'S SET THIS UP USING FUNCTION NOTATION,
AND THEN WE'LL FIND THE VALUE.
SO WE'D HAVE V = FOR THIS FIRST PARTITION
USING THE RIGHT ENDPOINT WE'D HAVE F OF 3,
THE HEIGHT OF THE RECTANGLE x DELTA X, THE WIDTH,
WHICH IS 2 UNITS PLUS LOOKING AT THE SECOND RECTANGLE
THE HEIGHT WOULD BE F OF 5 EVEN THOUGH THAT WOULD BE NEGATIVE.
THE WIDTH OR DELTA X WOULD BE 2 UNITS
+ LOOKING AT THE THIRD PARTITION,
AGAIN, WE'RE USING THE RIGHT SIDE,
SO WE CAN THINK OF THE HEIGHT AS F OF 8,
THOUGH IT WOULD BE NEGATIVE x DELTA X OR THE WIDTH
WHICH IS 3 UNITS.
NEXT, WE'LL DETERMINE THE FUNCTION VALUES, F OF 3,
F OF 5, AND F OF 8 USING F OF X = -X SQUARED + 16.
SO FOR F OF 3, SINCE 3 SQUARED IS 9 WE'D HAVE -9 + 16,
THAT'S 7.
SO WE HAVE 7 x 2 + FOR F OF 5,
SINCE 5 SQUARED = 25 THIS WOULD BE -25 + 16 WHICH WOULD BE -9.
SO WE HAVE -9 x 2 AND THEN FINALLY WE HAVE F OF 8 x 3.
F OF 8 WOULD BE -64 + 16.
THAT WOULD BE -48 x 3 WHICH MEANS WE HAVE 14
+ THIS WOULD BE -18 + THIS WOULD BE -144,
SO THE VALUE OF THE RIEMANN SUM WOULD BE -148.
I HOPE YOU FOUND THIS HELPFUL.