MATH 125 Antiderivatives and Areas WS 2.pdf - Net and Total Change 2 2 � 3(a Evaluate​ −2 |x2 − 4| dx and | �(x2 − 4)dx| and explain your

MATH 125 Antiderivatives and Areas WS 2.pdf - Net and Total...

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Net and Total Change 3 (a) Evaluate and and explain your answers. x | dx 2 −2 | 2 − 4 ( x ) dx | | 2 −2 2 − 4 for x and x x 2 − 4 > 0 < 2 > 2 for x 2 − 4 < 0 − 2 < x < 2 is always negative for (-2,2) x 2 − 4 x | dx x ) dx x x ) 2 −2 | 2 − 4 = 2 −2 ( 2 − 4 = ( 3 1 3 + 4 2 −2 = 3 16 + 3 16 = 3 32 ( x ) dx | x x ) | | 2 −2 2 − 4 = | 3 1 3 − 4 2 −2 | = | − 3 16 3 16 = 3 32 They are the same because x^2 -4 is negative for both (b) Now evaluate and and explain your answers. x | dx 3 −3 | 2 − 4 ( x ) dx | | 3 −3 2 − 4 for , x for , and x for 2 x 2 − 4 > 0 − 3 < x < 2 2 − 4 < 0 − 2 < x < 2 2 − 4 > 0 < x < 3 x | dx ( x ) dx x ) dx ( x ) dx 3 −3 | 2 − 4 = −2 −3 2 − 4 + 2 −2 ( 2 − 4 + 3 2 2 − 4 x x ) + − x x ) + x x ) ( 3 1 3 − 4 −2 −3 ( 3 1 3 + 4 2 −2 ( 3 1 3 − 4 3 2 ) ) ) ( 3 16 − 3 + ( 3 16 + 3 16 + ( 3 + 3 16 = 3 46 These are not the same because the x^2 - 4 changes signs throughout -3<x<3
Another Area Problem 4. An artist you know wants to make a figure consisting of the region between the curve and the x-axis for y = x 2 0 ≤ x ≤ 3 (i) Where should the artist divide the region with a vertical line so that each piece has the same area? (See the picture.) (ii)Where should the artist divide the region with vertical lines to get 3 pieces with equal areas? dx 3 x 2 = 9 (i) p is the value i am looking for -------> -------> dx p x 2 = 2 9 p 3 1 3 = 2 9 .38 p = 3 3 2 ≈ 2 (ii) q and r are the values I am looking for. dx and dx q p x 2 = 3 9 = 3 3 r x 2 = 3 The first integral is The second is q so q .08. 3 1 3 = 3 = √ 3 9 ≈ 2 r so r .62 9 − 3 1 3 = 3 = √ 3 18 ≈ 2

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