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Definite Integral

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

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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

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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 0 - 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)

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154 The Definite Integral 1. (a) Z - 1 - 2 f ( x ) dx = - 1 2 (b) Z 0 - 2 f ( x ) dx = Z - 1 - 2 f ( x ) dx + Z 0 - 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 0 - 1 ( | x - 2 | - x ) dx (b) Z 1 - 1 ( | x - 2 | - x ) dx 4. (a) Z 2 0 ( | x - 2 | - x ) dx (b) Z 5 1 ( | x - 2 | - x ) dx 5. (a) Z 5 1 ( | x - 2 | - x ) dx - Z 1 0 ( | x - 2 | - x ) dx (b) Z 1 0 ( | 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 0 n ( x ) dx (b) Z 0 - 2 n ( x ) dx - Z 4 0 n ( x ) dx + Z 6 4 n ( x ) dx (Do the three integrals separately.) 8. (a) Z 2 0 | x - 2 | dx - 2 Z 2 0 dx (b) Z 4 0 | x - 2 | dx + Z 0 4 2 dx 9. C ( x ) = p 1 - ( x - 1) 2 (a) Z 2 0 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 0 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 0 - 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 0 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 0 S ( x ) dx (b) Z 4 0 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 got your answer. If you use the fact that the function is even or odd you show this is true. 20. If f ( x ) is integrable on [ - 1 , 1], f ( x ) is an even function, and Z 1 0 f ( x ) dx = 1 3 Find Z 1 - 1 f ( x ) dx and Z 0 - 1 3 f ( x ) dx

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156 The Definite Integral 21. If f ( x ) is integrable on [ - π, π ], f ( x ) is an odd function and Z π 0 f ( x ) dx = 1 4 Find Z π - π f ( x ) dx 22. Z 2 0 | x | dx = 2, find Z 0 - 2 | x | dx and Z 2 - 2 | x | dx 23. Z - 2 2 x 3 1 - x 2 dx 24. Z 1 - 1 ( t 3 + t ) dt 25. Z π - π x 2 sin x dx 26. Z 4 - 4 x 15 cos x dx 27. Z π - π sin x cos x dx 28. Z π - π sin 2 x dx 29. If g ( x ) = 2 x - 4 evaluate Z 2 0 g ( x ) dx, Z 4 0 g ( x ) dx, Z 4 2 g ( x ) dx, Z 7 5 g ( x - 3) dx, 1 2 Z 8 4 g 1 2 dx 30. If f ( x ) = 3 x evaluate Z 1 0 ( f ( x ) - 3) dx, Z 3 1 ( f ( x ) - 3) dx, Z - 1 0 ( f ( x ) - 3) dx 31.
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