{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
2 CONTENTS
Background image of page 2
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)
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 4
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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Background image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}