– WELCOME TO TWO EXAMPLES INVOLVING THE ADDITION PROPERTY

OF DEFINITE INTEGRALS. TO HELP ILLUSTRATE

THESE EXAMPLES WE’LL ASSUME F OF X IS THIS

FUNCTION HERE, GRAPHED IN RED, WHICH IS A NONNEGATIVE FUNCTION. WE’RE GIVEN THE DEFINITE

INTEGRAL OF F OF X FROM 1 TO 6=THE DEFINITE INTEGRAL OF F

OF X FROM 1 TO 4 + THE DEFINITE INTEGRAL OF F

OF X FROM “A” TO B, AND WE’RE ASKED TO FIND

“A” AND B. BECAUSE F OF X IS NONNEGATIVE, IF WE INTEGRATE F OF X

FROM 1 TO 6 IT WOULD GIVE US THE AREA

OF THIS BLUE SHADED REGION, WHICH MEANS THE SUM OF THESE TWO

DEFINITE INTEGRALS MUST ALSO GIVE US THE SAME AREA. SO LOOKING AT THE RIGHT SIDE

OF THIS EQUATION WE HAVE THE INTEGRAL

OF F OF X FROM 1 TO 4. WELL, IF WE INTEGRATE

THIS FUNCTION ON THE INTERVAL FROM 1 TO 4,

IT WOULD GIVE US THIS AREA HERE. AND, THEREFORE,

TO GET THE TOTAL AREA, THIS SECOND INTEGRAL HERE

MUST BE INTEGRATED ON THE INTERVAL FROM 4 TO 6,

GIVING US THE REMAINING AREA, OR THIS AREA HERE. AND THAT MEANS “A” MUST BE EQUAL

TO 4, AND B MUST BE EQUAL TO 6. LET’S TAKE A LOOK

AT A SECOND EXAMPLE. HERE WE HAVE THE INTEGRAL

OF F OF X FROM 1 TO 3 MUST EQUAL THE DEFINITE INTEGRAL

OF F OF X FROM 1 TO 5 + THE DEFINITE INTEGRAL OF F

OF X FROM “A” TO B. AND ONCE AGAIN, WE WANT TO FIND

THE VALUE OF “A” AND B. SO NOTICE IN THIS CASE

ON THE LEFT SIDE, IF WE INTEGRATE FROM 1 TO 3 WE WOULD GET THIS AREA

HERE IN BLUE, SO WE WANT THE SUM OF THESE TWO

INTEGRALS TO BE EQUAL TO THIS BLUE AREA. BUT NOTICE THIS FIRST DEFINITE

INTEGRAL FROM 1 TO 5 WOULD GIVE US MORE AREA

THAN WE HAVE HERE IN BLUE. IT WOULD ACTUALLY GIVE US

ALL OF THIS AREA HERE, WHICH MEANS YOU NOW WANT TO

SUBTRACT OUT THIS AREA HERE. BUT NOTICE HOW WE DON’T HAVE

A SUBTRACTION SIGN HERE, WE HAVE AN ADDITION SIGN. SO IF WE WANT TO SUBTRACT

OUT THIS AREA HERE, INSTEAD OF INTEGRATING

FROM 3 TO 5, WE CAN CHANGE THE ORDER

OF INTEGRATION AND INTEGRATE FROM 5 TO 3. IF WE INTEGRATE F OF X

FROM 5 TO 3, IT WOULD RETURN

A NEGATIVE VALUE, OR WE CAN THINK OF IT AS

SUBTRACTING OUT THIS AREA HERE, WHICH MEANS “A”

WOULD BE EQUAL TO 5, AND B WOULD BE EQUAL TO 3. THAT’S GOING TO DO IT

FOR THESE TWO EXAMPLES. IN THE NEXT EXAMPLE WE’LL TAKE

A LOOK AT A PROBLEM WHERE WE HAVE A SUBTRACTION

OF TWO DEFINITE INTEGRALS.