5.4 Teorema Fundamental del Calculo

5.4 Teorema Fundamental del Calculo - Section 5.4 The...

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Section 5.4 The Fundamental Theorem of Calculus 315 {a, b}={0, }; n =10; dx = (b a)/n; 1 ± f = Sin[x] ; 2 xvals =Table[N[x], {x, a, b dx, dx}]; ± yvals = f /.x xvals; Ä boxes = MapThread[Line[{{#1,0},{#1, #3},{#2, #3},{#2, 0}]&,{xvals, xvals dx, yvals}]; ² Plot[f, {x, a, b}, Epilog boxes]; Ä Sum[yvals[[i]] dx, {i, 1, Length[yvals]}]//N Sums of rectangles evaluated at right-hand endpoints can be represented and evaluated by this set of commands. Clear[x, f, a, b, n] {a, b}={0, }; n =10; dx = (b a)/n; 1 ± f = Sin[x] ; 2 xvals =Table[N[x], {x, a dx, b, dx}]; ² yvals = f /.x xvals; Ä boxes = MapThread[Line[{{#1,0},{#1, #3},{#2, #3},{#2, 0}]&,{xvals dx,xvals, yvals}]; ± Plot[f, {x, a, b}, Epilog boxes]; Ä Sum[yvals[[i]] dx, {i, 1,Length[yvals]}]//N Sums of rectangles evaluated at midpoints can be represented and evaluated by this set of commands. Clear[x, f, a, b, n] {a, b}={0, }; n =10; dx = (b a)/n; 1 ± f = Sin[x] ; 2 xvals =Table[N[x], {x, a dx/2, b dx/2, dx}]; ²± yvals = f /.x xvals; Ä boxes = MapThread[Line[{{#1,0},{#1, #3},{#2, #3},{#2, 0}]&,{xvals dx/2, xvals dx/2, yvals}]; ±² Plot[f, {x, a, b},Epilog boxes]; Ä Sum[yvals[[i]] dx, {i, 1, Length[yvals]}]//N 5.4 THE FUNDAMENTAL THEOREM OF CALCULUS 1. (2x 5) dx x 5x 0 5(0) ( 2) 5( 2) 6 ' 2 0 ² o ² o ± ² ± o cd ab a b ## # ! ± # 2. 5 dx 5x 5(4) 5( 3) ' 3 4 ˆ‰ ’“ Š Š ± o ± o ± ± ± ± o xx 4 1 3 3 44 4 4 (3 ) # % ± $ ± 3. 3x dx 8 ' 0 4 Š Š Š ± o ± o ± ± ± o x3 x x 4 41 6 1 6 1 6 3(4) 3(0) (0) # % ! 4. x 2x 3 dx x 3x 2 3(2) ( 2) 3( 2) 12 ' 2 2 Š Š $# # # # ± # ± o o ±² ± ± ±²± o x2 44 4 (2 ) 5. x x dx x 0 1 ' 0 1 ˆ È #$ Î # " ! " ² o ² o ² ± o 2 33 6. x dx x (5) 0 2(5) 10 5 ' 0 5 $Î# &Î# &Î# $Î# & ! oo ± o o ± È 22 55 7. x dx 5x ( 5) ' 1 32 ± 'Î& ± "Î& $# " o ± o ± ±± o ± ‘ˆ 8. dx 2x dx 2x 1 '' 11 2 x1 ± o ± o ± # ± " ± " ± # ±± # ˆ‰ˆ‰
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316 Chapter 5 Integration 9. sin x dx [ cos x] ( cos ) ( cos 0) ( 1) ( 1) 2 ' 0 o ± o ± ±± o ±± ±± o 1 ! 1 10. (1 cos x) dx [x sin x] ( sin ) (0 sin 0) ' 0 ² o ² o ² ± ² o 1 ! 11 1 11. 2 sec x dx [2 tan x] 2 tan (2 tan 0) 2 3 0 2 3 ' 0 3 # Î$ ! oo ± o ± o 1 1 ˆ‰ ÈÈ 3 12. csc x dx [ cot x] cot cot 3 3 2 3 ' 6 56 # ' Î' o ± o ± o o 1 1 ˆ Š Š È 5 66 13. csc cot d [ csc ] csc csc 2 2 0 ' 4 34 )) ) ) o ± o ± o ± ±± o % Î% 1 1 ˆ Š 3 44 14. 4 sec u tan u du [4 sec u] 4 sec 4 sec 0 4(2) 4(1) 4 ' 0 3 ± o ± o 1 1 Î$ ! 3 15. dt cos 2t dt t sin 2t (0) sin 2(0) sin 2 '' 22 00 " ± " " " " " " " " ## # # # # ! Î# cos 2t 444 o ² o ² o ² ±² ± ‘ˆ ˆ 1 o ± 1 4 16. dt t sin 2t 33 " ² " " " " # # Î$ ² Î $ cos 2t 4 o ± o ± ± 1 1 sin 2 sin 2 sin sin o ± ± ± ± ± o ± ² ² o ± ˆ ˆ ˆ ‰ "" " " " " ² 1 1 1 1 1 1 1 3 3 6 43 6 4 3 3 È 17. 8y sin y dy cos y cos cos ' ab ’“ O ² O ² # Î# ² Î # ² ² o ± o ± ± o 8y 3 3 88 2 1 1 1 18. 4 sec t dt 4 sec t t dt 4 tan t ²² ² # ² Î % ² Î $ ± ² o ² o ± 1 1 tt 1 4 tan 4 tan (4( 1) 4) 4 3 3 4 3 3 o ± ± ± ± o ± ² ± ± ² o ± Š Š Š ˆ Š 19. (r 1) dr r 2r 1 dr r r ( 1) ( 1) 1 1 ² " ² " # # # ² " " ² ² o ² ² o ² ² o ² ± ² ± ± ² ² o ± Š Š r1 8 3 3 (1 ) 20. (t 1) t 4 dt t t 4t 2t 4t #$ # # $ ² $ o ² ² ² o ² ² ² a b È È 2 3 4 3 2 3 4 3 10 3 o ² ² ² ± ² ² ± ² ± o ³´ ³ ´ Š Š Š È È È Š Š Š Š È È 3 3 4 3 21. du u du " " ² & " # Š Š Š uu u 1 3 u1 6 1 6 1 6 4 4u 4(1) 2 42 ± o ± o ² o ² ± ² o ± È Š È Š È 22.
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This note was uploaded on 03/19/2012 for the course ELECTRIC 311 taught by Professor Rene during the Spring '12 term at Uni San Francisco de Quito.

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5.4 Teorema Fundamental del Calculo - Section 5.4 The...

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