Calculus I - Module 10 Definite Integrals

16.1 Definite integrals Adefinite integral of F from a to b is denoted by S Fix dx the number othe a is lower limit b function f x is upper limit is calledthe integrand and x is the variable of integration which represented by dx Theorem 1 Area flax betArea FCXldx If X is t Y tag or Area above x axis Area below x axis For How to Solve On Graph Go Page 318 T to h Evaluating using geometric features a g text'd I B f 11 7 z Area I8 2x 6 t 18 dx 241 6 8 21 5 6 4 Evaluate 1 Using Definite integrals let fog a b J fix dx J af be integral functions on interval that contains s c let K a b B be real 0 2.1 flxSdx J faddy 3 Lik fix dx Kf Fade 4 5 f x dx Baix af f xldxIBf giddy fly dx fflxldxtffixsdxac.ca b 6 the function Al integrable and Jal fallax is the sum Of areas of theregion bounded by graph off and the X axis on La b Evaluate a f b f C d f Assumes flxdx 3 21 fax 2fLxldx I fix dx fo fixidx 3 fix dx f 10 21 102 fix dx 6 fix dx tf Fxldx f fix dx 6 fax dx First gypeffit to 21 10 3 Es 3 115 12 6 flax 3 640 60 Even odd function b A function is even if it Satifies property FL this means its symmetric Ex F A function x x2 F is Odd if X across if f is F X x f XI P FA cosy f x FL x fax that means Origin x3 f F xldx X x fix dx 2 even odd x axis x Its Symmetric across Ex F y Fox x o z fix sine tax fix dx 0 Evaluate Suppose Flax 7,1 flxidx 4,1guides Iguidx and P s a flxldx Even b 1,91 14 5 odd c S flax is odd g is even ffexldx.SE FIX dX 171 2 891 1dx faux 2167 f G faldex C4 L 11 1 0 1 10.2 If f Fundamental Theorem Of Calculus J flax FLDS Fla 1 Find Antiderivative F J and F is anti derivative a b is continues on x Fox dx Dont write integration C F x 2 Substitute integration limit b b a FIX dx LFLx roux in Feb FG Fla Evaluate a f x 3 3 7 dx 18 b f It 364 L A dx t 34 3 3 3 43 t 81z 32 4314,32 Y F 3320 i using fundamental theorem to find Area Find area of region bounded by x2 2 Y is b Vertive X the ax 1 Area f x 3 214 4 3 lines x2 2 3 3 45 15 to 105 34 3 o region bounded by graph y 3A the x 1 Area SIZE 2 3 Ex 2 X 4 2 56 P 3 41 pindsalenitF 4 Zo 3 Pverticle axis 3 39 11259mg x y 21167 47 2143 47 6 x 64 8 56 U 6 3 du 0 36 5 I 4 ut 5 5 te 6 6 Etc The Second theorem If f calculus of is continuous on an open interval then for every x If Fladt if U vex a is I function the in of X then Fluexis u'd Fade Isin't it S b SITE FI C Fosceldt Cos 1 2 u 2x interval FA Evaluate a I containing 2xco C Mean The t average for integrate value Mean value flx dx Ffc the flat given as Average F Evaluate FIX1 in value 5 of F Ia f f on 1 3 theorem is interval a b 2x Value interval of O I 2 7 E E 3111 known dx Fixidx 2 2 on Average 2 11 x a valve mean affix Find b o of function E Evaluate FIX Ia f F on i e Fixldx f s loenixi et loenlel Wen lil 21 41 XCX 1 ICE 5,8 Ix Ix x2 5118 s 4136 51313 s 2 10.3 Integration by let U g x Where g let f substitution is continuous on be continuous ffCgcx g x on range a of b and y Fluid dx Procedure 2 Compute differential of U du glexidx 3 Rewrite integral in terms of variable u 4 Evaluate new indefinite pattern Flyer g Recognize X2 A 1 2 du 3 1 2x dx x2 I U integral we respect to gex FIX g x x inner IS X is 2x 2xdx x 1 2x dx 4 fu du Etc v3 5 x2 C I PTC u b Apply x2 1 u b 27 dx 3 cos 3 3 U du 2 gex FE 3 Gsx 3dx S3cos 2 dx 3 ICosudu Sino te C Josie s'dx U du X ÉÉÉ É 5 3x2dx 13 2 e dx 2 e'du cute elx3tsy C When a is fax U n'do 1 XP we 941151 I X du missing 2x 1 2 1 dx fudu x2 1 L I 2x te xt_