Ex: Property of Definite Integral Subtraction

Ex: Property of Definite Integral Subtraction


– WELCOME TO AN EXAMPLE
OF THE SUBTRACTION PROPERTY OF DEFINITE INTEGRALS. TO HELP ILLUSTRATE
THIS PROPERTY, WE’LL ASSUME F OF X IS GRAPHED
HERE IN RED AND IS NONNEGATIVE. BUT F OF X DOES NOT NEED
TO BE NONNEGATIVE FOR THIS PROPERTY TO HOLD TRUE. WE HAVE THE DEFINITE INTEGRAL
OF F OF X FROM 2 TO 9 – THE DEFINITE
INTEGRAL OF F OF X FROM 2 TO 4=THE DEFINITE INTEGRAL OF F
OF X FROM “A” TO B, AND WE’RE ASKED TO FIND
THE VALUE OF “A” AND B. SO, AGAIN, ASSUMING F OF X
IS NONNEGATIVE, IF WE INTEGRATE F OF X ON THE
CLOSED INTERVAL FROM 2 TO 9, IT WOULD GIVE US THE AREA
OF THIS BLUE SHADED REGION. THEN IF WE SUBTRACT THE DEFINITE
INTEGRAL F OF X ON THE INTERVAL FROM 2 TO 4, WE WOULD BE SUBTRACTING
THIS AREA HERE, LEAVING US WITH THE AREA ON THE
CLOSED INTERVAL FROM 4 TO 9. AND, THEREFORE, THE DIFFERENCE
OF THESE TWO DEFINITE INTEGRALS MUST BE EQUAL TO THE REMAINING
BLUE AREA, AND, THEREFORE, “A”
WOULD BE 4 AND B WOULD BE 9.  

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