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X
- P PRIME OF X = 20X IS THE MARGINAL PROFIT FUNCTION
WHERE P PRIME OF X IS IN THOUSANDS OF DOLLARS
AND X IS IN MONTHS.
WE WANT TO DETERMINE THE AREA UNDER THE GRAPH
BOUNDED BY THE X-AXIS ON THE CLOSED INTERVAL FROM 0 TO 3
AND THEN INTERPRET THE RESULTS.
SO HERE IS THE GRAPH OF OUR DERIVATIVE FUNCTION.
WE WANT TO DETERMINE THE AREA UNDER THIS GRAPH
ABOVE THE X-AXIS ON THE INTERVAL FROM 0 TO 3,
SO WE WANT TO DETERMINE THE AREA OF THIS TRIANGULAR REGION HERE.
AND SINCE THIS REGION IS A TRIANGLE
WE CAN USE THE AREA FORMULA FOR A TRIANGLE
TO DETERMINE THIS AREA
WHERE THE BASE IN THIS CASE WOULD BE 3 UNITS,
AND THE HEIGHT OF THE TRIANGLE, SINCE THIS IS A RIGHT TRIANGLE,
WOULD BE THE Y-COORDINATE OF THIS POINT OR 60 UNITS.
BUT BECAUSE THIS FUNCTION IS CONTINUOUS AND NON-NEGATIVE
ON THIS INTERVAL
WE CAN REPRESENT THIS AREA AS A DEFINITE INTEGRAL.
THE AREA WOULD BE EQUAL TO THE DEFINITE INTEGRAL FROM 0 TO 3
OF OUR FUNCTION 20X WITH RESPECTS TO X.
BUT NOW TO DETERMINE THIS AREA
WE WILL USE THE GEOMETRIC FORMULA OF 1/2 x BASE x HEIGHT,
SO WE'LL HAVE 1/2 x 3 x 60 WHICH IS EQUAL TO 90.
AND NOW TO HELP US UNDERSTAND WHAT THIS MEANS,
NOTICE THAT WE MULTIPLIED THE PROFIT PER MONTH
IN THOUSANDS OF DOLLARS BY THE NUMBER OF MONTHS,
SO THIS 90 IS NOT ONLY THE AREA OF THIS TRIANGLE,
IT ALSO REPRESENTS THE THOUSANDS OF DOLLARS OF PROFIT
FOR THE FIRST THREE MONTHS,
SO THE PROFIT FOR THE FIRST THREE MONTHS WOULD BE $90,000
BASED UPON THIS SITUATION.
NOW, IN THIS EXAMPLE IT WAS PRETTY EASY TO DETERMINE
THE AREA UNDER THE CURVE
BECAUSE OF THE GEOMETRIC FORMULA FOR THE AREA OF A TRIANGLE,
BUT THIS IS NOT ALWAYS GOING TO BE THE CASE.
IF WE DON'T HAVE A GEOMETRIC FORMULA
WE'RE GOING TO HAVE TO USE RECTANGLES
TO APPROXIMATE THE AREA UNDER THE CURVE,
AND THEN LATER ON WE'LL LEARN HOW WE CAN USE ANTIDERIVATIVES
TO DETERMINE AREA UNDER FUNCTIONS
USING THE FUNDAMENTAL THEOREM OF CALCULUS.
BUT WE WILL TAKE A LOOK AT ONE MORE EXAMPLE
OF DETERMINING AREA USING A GEOMETRIC FORMULA.
I HOPE YOU FOUND THIS HELPFUL.