lecture16 - Lecture 16 - The Area Between Two Functions...

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Unformatted text preview: Lecture 16 - The Area Between Two Functions 16.1 The Area Between Two Curves We can use the definite integral to calculate the area between two curves. Its easy to see visually that if f and g are both positive and f ( x ) g ( x ) for all x in [ a,b ], then the area between the two graphs between a and b is Z b a f ( x ) dx- Z b a g ( x ) dx For this to be true we need that f ( x ) g ( x ) to hold. Examples: 1. Find the area bounded by y = x and y =- x on the interval [1 , 4]. solution: In all of these area problems you should begin by sketching the curve Here x - x for all x in the interval, so the area is Z 4 1 ( x- (- x )) d x = x 3 / 2 3 / 2 + x 2 2 4 1 = 2 3 (8) + 16 2- 2 3 + 1 2 = 73 6 2. Find the area between the curves y = x 2 + 3 and y =- x 2- 1 on [- 2 , 2] solution: x 2 + 3 - x 2- 1 everywhere and therefore certainly on [- 2 , 2]. Thus the area is Z 2- 2 (( x 2 + 3)- (- x 2- 1)) d x = Z 2- 2 (2 x 2 + 4) d x = 2 x 3 3 + 4 x 2- 2 = 2 3 (8) + 8- 2 3 (- 8)- 8 = 80 3 3. Find the area bounded by y = 2- x 2 and y = x ....
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lecture16 - Lecture 16 - The Area Between Two Functions...

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