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- IN THIS LAST EXAMPLE WE WANT TO APPROXIMATE THE AREA
UNDER THE SAME FUNCTION F OF X = 4
DIVIDED BY X ON THE SAME CLOSED INTERVAL FROM 1 TO 5
BUT NOW WE WANT TO USE 8 RIGHT-SIDED RECTANGLES.
SO AGAIN WE'LL START BY MARKING OFF THE INTERVAL FROM 1 TO 5,
SO WE'RE GOING TO START HERE AND STOP HERE
AND NOW WE WANT TO DIVIDE THIS INTO 8 EQUAL PARTS.
IF WE WANTED TO WE COULD USE THIS FORMULA HERE
TO DETERMINE THE WIDTH OF EACH INTERVAL LABELED DELTA X
WHERE A WOULD BE 1 AND B WOULD BE 5 AND N WOULD BE 8.
SO LET'S GO AHEAD AND USE THE FORMULA THIS TIME,
DELTA X WOULD BE = 5 - 1 DIVIDED BY 8
WHICH WOULD BE 4/8 OVER 1/2,
SO THIS TELL US THE WIDTH OF EACH INTERVAL
IT WOULD BE 1/2 OF A UNIT.
SO THERE ARE THE 8 = INTERVALS AND NOW WE WANT TO USE
THE RIGHT-SIDE OF EACH INTERVAL
TO DETERMINE THE RECTANGLE'S HEIGHT.
SO LOOKING AT THIS FIRST INTERVAL
WE'LL USE F OF 1 1/2 OR F OF 3/2
FOR THE HEIGHT OF THIS FIRST RECTANGLE.
FOR THE SECOND WE'LL USE F OF 2,
FOR THE 3RD WE'LL USE F OF 2 1/2 OR F OF 5/2.
FOR THE 4TH WE'LL USE F OF 3.
FOR THE 5TH WE'LL USE F OF 3 1/2 OR F OF 7/2.
FOR THE 6TH WE'LL USE F OF 4.
FOR THE 7TH WE'LL USE F OF 4 1/2 OR F OF 9/2
AND FOR THE LAST RECTANGLE WE'LL HAVE A HEIGHT OF F OF 5.
NOTICE HOW THE SUM OF THE AREAS OF THESE 8 RECTANGLES
WOULD BE LESS THAN THE AREA UNDER THE BLUE FUNCTION,
SO THIS IS OFTEN CALLED THE LOWER SUM OF THE RECTANGLES.
AND AGAIN HERE THE IDEA HERE IS THE AREA
UNDER THE FUNCTION IS GOING TO BE APPROXIMATELY
EQUAL TO THE SUM OF THE AREA OF THESE 8 RECTANGLES,
SO WE'LL SAY 8 SUB 1 + A SUB 2 ALL THE WAY THROUGH A SUB 8.
SO IN GENERAL THE AREA IS GOING TO BE APPROXIMATELY
EQUAL TO THE SUM OF F OF C SUB I X DELTA X WHERE I GOES FROM 1
TO N AND N REPRESENTS A NUMBER OF A RECTANGLES.
SO FOR THIS SITUATION THE AREA IS GOING TO BE APPROXIMATELY
EQUAL TO THE SUM OF THESE 8 AREAS
WILL HAVE F OF 1 1/2 OR F OF 3 1/2 X THE WIDTH OF 1/2.
AGAIN, THIS IS THE HEIGHT X THE WIDTH PLUS FOR THE 2ND RECTANGLE
WILL HAVE F OF 2 x 1/2 FOR THE 3RD
WE'LL HAVE F OF 2 1/2 OR F OF 5/2 x 1/2
+ THE 4TH RECTANGLE HAS A HEIGHT OF F OF 3 x THE WIDTH OF 1/2
+ THE 5TH HAS A HEIGHT OF F OF 3 1/2
OR F OF 7/2 AND THE WIDTH OF 1/2
+ THE 6TH HAS A HEIGHT OF F OF 4.
THE 7TH HAS A HEIGHT OF F OF 4 1/2 OR F OF 9/2 x 1/2
+ THE LAST RECTANGLE HAS A HEIGHT OF F OF 5 x 1/2.
SO IF WE GO BACK TO THE PREVIOUS VIDEO JUST FOR A MOMENT
AND COMPARE HOW WE'RE DETERMINING THESE AREAS,
IT'S ALMOST THE SAME EXCEPT NOTICE
THAT FOR THE RIGHT SIDED RECTANGLES
WE SUB IN AN X VALUE OF 3/2 IN THE FUNCTION
TO DETERMINE THE HEIGHT AND THEN WE GO ALL THE WAY TO F OF 5.
BUT FOR THE LEFT-SIDED RECTANGLES IN THE PREVIOUS VIDEO
WE DETERMINED THE FUNCTION VALUE X = 1
AND WENT ALL THE WAY FROM 1 TO 9/2.
SO I THINK IT'S REALLY IMPORTANT TO SKETCH THE RECTANGLES
TO DETERMINE EXACTLY HOW TO SET THESE UP.
NOW, FROM HERE WE'RE GOING TO GO AHEAD AND EVALUATE THIS
IN A CALCULATOR.
SO IN ORDER TO HAVE THE FUNCTION IN Y1
AND THEN FROM THE HOME SCREEN WE CAN JUST TYPE IN ALL OF THIS
AT ONE TIME.
INSTEAD OF USING F WE'LL USE Y OF 1.
SO WE'LL PRESS VARS, RIGHT ARROW, ENTER, ENTER
AND THEN IN PARENTHESIS WE HAVE 3/2 FOR F OF 3/2 x 1/2 + VARS,
RIGHT ARROW, ENTER, ENTER AND THEN WE HAVE F OF 2.
SO WE'LL HAVE Y 1 OF 2 x 1/2
AND NOW I'M GONNA GO AHEAD AND PAUSE THE VIDEO
AND TYPE THE REST IN.
OKAY, SO NOW I THINK I HAVE ALL THE INFORMATION IN THERE.
NOW WE CAN PRESS ENTER.
IT GIVES YOU DECIMAL APPROXIMATION
BUT IF WE PRESS MATH, ENTER,
ENTER WE CAN DETERMINE THE FRACTION VALUE.
SO IT'S 3,601 ALL OVER 630.
SO THE APPROXIMATE AREA UNDER THE FUNCTION IS 3,601
DIVIDED BY 630 SQUARE UNITS.
AND THE ACTUAL AREA WILL BE A LITTLE BIT MORE THAN THIS
BECAUSE THIS IS THE LOWER SUM
AND FROM THE PREVIOUS VIDEO THE UPPER SUM WAS 4,609
DIVIDED BY 360 WHICH MEANS THE AREA UNDER THE FUNCTION
WOULD ACTUALLY BE SOMEWHERE IN BETWEEN THESE TWO VALUES.
BUT USING 8 RIGHT-SIDED RECTANGLES
THIS WOULD BE OUR APPROXIMATION FOR THE AREA UNDER THE FUNCTION.
I HOPE YOU FOUND THESE EXAMPLES HELPFUL.