Definite Integral - Contents 10 The Definite Integral 153...

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Unformatted text preview: Contents 10 The Definite Integral 153 10.1 The Definite Integral and Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 10.2 Integrals with Variable Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 10.3 Derivative of an Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 10.4 Integrals of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 10.5 Evaluation of Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 2 CONTENTS Chapter 10 The Definite Integral 10.1 The Definite Integral and Area In problems 1 to 19 do the following: (i) Graph the function. (ii) Shade in the area between the function and the interval of integration on the x-axis with plus or minus signs depending whether the given area is above and/or below the x-axis. Draw a separate diagram for each interval of integration. (iii) Compute the definite integral. 1. f ( x ) = x + 1 (a) Z- 1- 2 f ( x ) dx (b) Z- 2 f ( d ) dx Solution for 1: y x y x + f ( x ) = x + 1 f ( x ) = x + 1 +++ (a) (b) +++++ (- 2 , 0) (- 1 , 0) (- 2 , 0) (- 1 , 0) 154 The Definite Integral 1. (a) Z- 1- 2 f ( x ) dx =- 1 2 (b) Z- 2 f ( x ) dx = Z- 1- 2 f ( x ) dx + Z- 1 f ( x ) dx =- 1 2 + 1 2 = 0 2. f ( x ) = x + 1. (a) Z 9- 1 f ( x ) dx (b) Z 1- 1 f ( x ) dx (c) Z 1- 2 f ( x ) dx 3. (a) Z- 1 ( | x- 2 | - x ) dx (b) Z 1- 1 ( | x- 2 | - x ) dx 4. (a) Z 2 ( | x- 2 | - x ) dx (b) Z 5 1 ( | x- 2 | - x ) dx 5. (a) Z 5 1 ( | x- 2 | - x ) dx- Z 1 ( | x- 2 | - x ) dx (b) Z 1 ( | x- 2 | - x )- Z 5 1 ( | x- 2 | - x ) dx 6. Let h ( x ) = | x- 2 | - 2 (a) Z 2- 2 h ( x ) dx (b) Z 6- 2 h ( x ) dx 7. Let n ( x ) = | x- 2 | - 2 (a) Z 4 n ( x ) dx (b) Z- 2 n ( x ) dx- Z 4 n ( x ) dx + Z 6 4 n ( x ) dx (Do the three integrals separately.) 8. (a) Z 2 | x- 2 | dx- 2 Z 2 dx (b) Z 4 | x- 2 | dx + Z 4 2 dx 9. C ( x ) = p 1- ( x- 1) 2 (a) Z 2 C ( x ) dx (b) Z 2 1 C ( x ) dx 10. G ( x ) = ( p 1- ( x- 1) 2 , x ∈ [0 , 2]- p 1- ( x + 1) 2 , x ∈ [- 2 , 0) (a) Z 2- 2 G ( x ) dx (b) Z 2- 1 G ( x ) dx 11. Let F ( x ) = x (a) Z 2 F ( x ) dx (b) Z 4 2 F ( x- 2) dx 10.1 The Definite Integral and Area 155 12. Let A ( x ) = | x | (a) Z- 1 A ( x ) dx (b) Z 10 9 A ( x- 10) dx 13. Let f ( x ) = x + 1 (a) Z- 1- 2 f ( x ) dx (b) Z 3 2 f ( x- 4) dx 14. Let L ( x ) =- x + 2 (a) Z 4 2 L ( x ) dx (b) Z 7 5 L ( x- 3) dx 15. Let L ( x ) =- x + 2 (a) 1 2 Z 2 1 L x 2 dx (b) 4 Z 1 1 2 L (4 x ) dx 16. Let A ( x ) = | x | (a) Z 6- 3 A ( x ) dx (b) 3 Z 2- 1 A (3 x ) dx 17. Let f ( x ) = 1 2 . Compute the following: Z 2 1 f ( x ) dx, Z 2 1 4 f ( x ) dx, Z- 4- 3 f ( x ) dx, Z 42 41 f ( x- 40) dx, Z 24 23 f ( x + 20) dx 18. Let g ( x ) = 2 x- 4 (a) Z 2 g ( x ) dx (b) Z 4 2 g ( x ) dx 19. Let S ( x ) =    3 x 4 , x ∈ [0 , 4] √ 25- x 2 , x ∈ (4 , 5] (a) Z 5 S ( x ) dx (b) Z 4 S ( x ) dx (c) Z 5 4 S ( x ) dx In problems 20 to 39 give the value of the definite integral and explain in words and/or a diagram how you...
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This note was uploaded on 09/22/2011 for the course ECON 101 taught by Professor Mr.tull during the Spring '11 term at De La Salle University.

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Definite Integral - Contents 10 The Definite Integral 153...

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