The Petronas Towers of Kuala Lumpur
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1 BA0 ENGINEERING MATHEMATICS 0 CHAPTER 4 INTEGRATION 4. INTRODUCTION TO INTEGRATION Why do we need to study Integration? The Petronas Towers of Kuala Lumpur Often we know the relationship involving the rate of change of two variables, but we may need to know the direct relationship between the two variables. For example, we may know the velocity of an object at a particular time, but we may want to know the position of the object at that time. To find this direct relationship, we need to use the process which is opposite to differentiation. This is called integration (or antidifferentiation). The processes of integration are used in many applications. The Petronas Towers in Kuala Lumpur experience high forces due to winds. Integration was used to design the building for strength. 87
2 BA0 ENGINEERING MATHEMATICS 0 Sydney Opera House The Sydney Opera House is a very unusual design based on slices out of a ball. Many differential equations (one type of integration) were solved in the design of this building. Wine cask Historically, one of the first uses of integration was in finding the volumes of wine-casks (which have a curved surface). Other uses of integration include finding areas under curved surfaces, centres of mass, displacement and velocity, fluid flow, modelling the behaviour of objects under stress, etc. 88
3 BA0 ENGINEERING MATHEMATICS Antiderivatives and The Indefinite Integral Integration is the inverse process of differentiation. d x x c 4x 4x dy f ' x C is arbitrary constant ' f x y c integrand integral The term identifies x as the variable of integration Two types of integrals Indefinite integral Definite integral 89
4 BA0 ENGINEERING MATHEMATICS 0 4. Indefinite Integral 4... Integration Of Constants n ax n n ax ax c, n ( power rule) n Examples Evaluate the following equation; i. 4x ii. dp iii. s ds iv. x 90
5 BA0 ENGINEERING MATHEMATICS 0 Exercise a) b) c) 4 d) x e) x f) x g) x h) 8 x i) 0 x 4... Integration Of Summation & Subtraction ax n bx m ax n bx m n m ax bx n m c 9
6 BA0 ENGINEERING MATHEMATICS 0 Example Find the derivative for each of the following functions. x x b) 6 x 9 x x c) a) t 4 dt t d) x e) x x x f) p p dp g) 4 x 4 x h) x x i) x4 x 4 x 9
7 BA0 ENGINEERING MATHEMATICS INTEGRATION BY SUBSTITUTION n ax b, n Step : Step : Let u Differentiate u Step : Make as a subject Step 4: Integrate the u equation n n du ax b u a n u c n a u n Step 5 : Substitute back u So the final answer is n ax n c a b n a Examples: Evaluate the following equation: Find 4 5 x 9
8 BA0 ENGINEERING MATHEMATICS 0 Example Find the following integration by substitution: 5 a). ( x 7) b). (5x 8) 4 C). x d). ( 6 x 7) 94
9 BA0 ENGINEERING MATHEMATICS INTEGRATION OF OR LOGARITHM FUNCTION x x, n=- x ln x d x c ln ax b c ax b a Example : i. 4x ii. 4 9x iii. t dt iv. 8 6 x x 95
10 BA0 ENGINEERING MATHEMATICS 0 Exercise: a) 5x b) x x c) d) x x x INTEGRATION OF EXPONENTIAL FUNCTION x x e e c ax ax e e c a e d ax b axb axb e c 96
11 BA0 ENGINEERING MATHEMATICS 0 Example: a) 5x e b) x e c) 4 e x d) e x t e) e t dt f) e x e x INTEGRATION OF TRIGONOMETRIC FUNCTION cos ax b sin ax b c d ax b sin ax b cosax b c d ax b tan ax b sec ax b c d ax b 97
12 BA0 ENGINEERING MATHEMATICS 0 sin ax cosax c a cosax sin ax c a sec ax tan ax c a Example: a) sin x b) cos 4 sec x x c) sin x d) sec x e) cos x sin x f) 98
13 BA0 ENGINEERING MATHEMATICS 0 Challenge:. p dp p p. 7. tan x 4. k k dk 5e e x x 99
14 BA0 ENGINEERING MATHEMATICS 0 Solve the integration below by using substitution: a). 4x 6 b). 5 c) x 4 x x e 4 x d). 5 x e e). sin x cos x f). 4 x sin x 00
15 BA0 ENGINEERING MATHEMATICS 0 Integrate for Example: sin mx and cos mx sin cos mx mx cos cos mx mx a) sin x b) cos x 0
16 BA0 ENGINEERING MATHEMATICS 0 x c) 4sin 4. DEFINITE INTEGRAL 4.. Definite Integral Examples: If f x F x c b So f x F b F a 0 a. 5 0
17 BA0 ENGINEERING MATHEMATICS 0. ( x ). 4 e 4x 4. 4 sec z dz 0
18 BA0 ENGINEERING MATHEMATICS x 4 Exercise: x x 4 x x 4. x x 04
19 BA0 ENGINEERING MATHEMATICS DEFINITE INTEGRAL BY SUBSTITUTION METHOD. Find the value of 4 a) 4x b) x 6x4 0 x e 05
20 BA0 ENGINEERING MATHEMATICS 0 c) 4 sec x 0 d) 0 e 0.t dt 06
21 BA0 ENGINEERING MATHEMATICS 0 Exercise:. 4x. xx. x 0 x 5 07
22 BA0 ENGINEERING MATHEMATICS PROPERTIES OF THE DEFINITE INTEGRAL b a) a kf x k f x b a b b b b) f x f x f x f x a a a b c) a f x f x a b c c d) Examples: a b a b f x f x f x 6 i. Given that 6 a) f f x 4, find the value of: x 6 b) f x 08
23 BA0 ENGINEERING MATHEMATICS 0 c) 6 f x 4 6 d) f x f x 4 e) k if 6 f x kx 09
24 BA0 ENGINEERING MATHEMATICS 0 Exercise: Given that f x 4 and f x 6, evaluate each of following a) f x b) 5 f x c) f x d) 4f x 0
25 BA0 ENGINEERING MATHEMATICS 0 POLITEKNIK KOTA BHARU JABATAN MATEMATIK, SAINS DAN KOMPUTER BA 0 ENGINEERING MATHEMATICS PAST YEAR FINAL EXAMINATION QUESTIONS JULAI 00 Integrate this following: i. ii. ( ) iii. iv. ( ) v. vi. ( ) JANUARI 00 a) Integrate this following: i. ( ) ( ) ii. iii. b) Evaluate the integrals: i. ( ) ii. ( ) JULAI 009 Integrate: a) ( ) b) c) ( ) d) ( ) e) ( )( ) JANUARI 009 a) Solve the following of indefinite integrals: i. ii. iii. iv. ( ) b) Solve the following of definite integrals: i. ( ) ii. ( ) JULAI 008 Integrate: i. ii. iii. ( ) iv. ( )( ) v. ( )
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