Chapter 3(A) - Chapter 3 Integration Overview n Integral q...

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Chapter 3 Integration
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Overview n Integral q Indefinite Integral q Definite Integral n Fundamental Theorem of Calculus n Various Integration Techniques q Integration by Substitution q Integration by Parts
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Overview n Application of Integration q Area between two curves q Volume of Solids of Revolution
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Integrals
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Indefinite Integral 2 Let ( )3. f xx = 23 Then 3 x d xxC =+ We call 3 xC + or 2 3 x dx 2 indefinite integral the o 3 f x The of .. ( ) f wrtx fx dx = indefinite integral
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We call 3 xC + or 2 3 x dx 2 indefinite integral the o 3 f x If we a value to , we ge antiderivative t an of ) fx . i ( C 3 1 x + 3 2 x + 3 3 x + 3 a fix number x + 2 antiderivatives ( 3 of ) . f xx =
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If we a value to , we ge antiderivative t an of ) fx . i ( C 3 1 x + 3 2 x + 3 3 x + 3 a fix number x + 2 antiderivatives ( 3 of ) . f xx = Note : 32 ( 1 )3 d dx += ( 2 d dx ( a fix number d dx ( 3 d dx antiderivativ If we differentiate of , the answer i e () ( s . ) f x
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Procedure Reverse ) ( ) ( ' ) ( ation Differenti x f x F x F = if interval an on function a of an called is function A I f F tive antideriva ' ( ) ( ) for all F x f x xI =∈ antiderivativ If we differentiate of , the answer i e () ( s . ) f fx x
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Indefinite Integral The of indef inite in .. tegra ( ) l f wrtx fx = the set of all of antiderivatives f =
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Indefinite Integral - Remark . at ) ( slopes their have which ) ( curves all find to is on process the of tion interpreta l geometrica The x x f C x F y + = n integratio x f ( x ) = 2 x y x 2 -3 1 0 1 2 + x 3 2 - x
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Indefinite Integral If is an of on , then is also an of on and every of on is of this form. F f I FC f If I + antiderivative antiderivativ e antiderivative C x F dx x f + = ) ( ) ( Integral sign Integrand Constant of integration
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If for all (, ), then there exists such that () ' for all (,) ) ') ( ( G xF x xC F ab xa x b x GC =+ = Example: 3 ( ) 2009 F xx 3 ( )1 G 2 ' ( )3 F = 2 ' ( G = Therefore, ' ( ) ' ( ). F x Gx = Note : 3 3 ( ) 2009 1 2008 ( ) 2008 F x Gx = ++
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If for all (, ), then there exists such that () ' for all (,) ) ') ( ( G xF x xC F ab xa x b x GC =+ = Pause and Think !!! How to prove trigonometric identities using the above result? 22 si n co s1 1 ta n sec co t 1 csc xx +=
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If for all (, ), then there exists such that () ' for all (,) ) ') ( ( G xF x xC F ab xa x b x GC =+ = 22 To prove : si n co s1 xx += Let 2 ( ) sin F = 2 ( ) cos G =- Then ' ( ) 2si n cos F x = ' ( ) 2co s( si n) n cos G x = -- = Thus ' ( ) ' F x Gx = Hence ( ) F x G Therefore si n cos x = -+ Question: How to find C ??
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22 To prove : si n co s1 xx += Let 2 ( ) sin F = 2 ( ) cos G =- Then ' ( ) 2si n cos F x = ' ( ) 2co s( si n) n cos G x = -- = Thus ' ( ) ' () F x Gx = Hence ( ) F x G xC =+ Therefore si n cos x = -+ Question: How to find C ??
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This note was uploaded on 11/01/2011 for the course MATH 1505 taught by Professor Yap during the Spring '11 term at National University of Singapore.

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Chapter 3(A) - Chapter 3 Integration Overview n Integral q...

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