5.6 Area entre curvas

# 5.6 Area entre curvas - Section 5.6 Substitution and Area...

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Section 5.6 Substitution and Area Between Curves 329 s 8t cos ( t) 1 s(1) 8 cos 1 10 m Êo ± ² Ê o ± ² o 11 61. All three integrations are correct. In each case, the derivative of the function on the right is the integrand on the left, and each formula has an arbitrary constant for generating the remaining antiderivatives. Moreover, sin x C 1 cos x C C 1 C ; also cos x C C C C . ## # "" # " # # \$ # " # # " ² o ± ² Ê o ² ± ² o ± ± ² Ê o ± o ² cos 2x 62. Both integrations are correct. In each case, the derivative of the function on the right is the integrand on the left, and each formula has an arbitrary constant for generating the remaining antiderivatives. Moreover, C C C tan x sec x 1 sec x ### # ± " ² o ² o ² ± ð ñò ˆ‰ a constant 63. (a) V sin 120 t dt 60 V cos (120 t) [cos 2 cos 0] Š ± ± "Î'! ! # 60 0 120 V ' 0 16 0 max max 1 o ± o ± ± max [ 1 1 ] 0 o ± ± o V max # 1 (b) V 2 V 2 (240) 339 volts max rms oo ¸ ÈÈ (c) V sin 120 t dt V dt (1 cos 240 t) dt '' ' 00 0 0 0 0 max max ab # ± ± 1c o s 2 4 0 t V 1 max t sin 240 t sin (4 ) 0 sin (0) o ± o ± ± ± o VV V 240 60 240 40 1 0 max max max # # " " "Î'! ! ± ± ˆ ˆ 1 5.6 SUBSTITUTION AND AREA BETWEEN CURVES 1. (a) Let u y 1 du dy; y 0 u 1, y 3 u 4 o ² o Ê o o Ê o y 1 dy u du u (4) (1) (8) (1) 01 34 È ± ‘ˆ ˆ ˆ ˆ ² o o o ± o ± o "Î# \$Î# \$Î# \$Î# % " 222 2 2 1 4 333 3 3 3 (b) Use the same substitution for u as in part (a); y 1 u 0, y 0 u 1 o ± o Ê o y 1 dy u du u (1) 0 10 È ± ² o o o ± o "Î# \$Î# \$Î# " ! 22 2 33 3 2. (a) Let u 1 r du 2r dr du r dr; r 0 u 1, r 1 u 0 o ± ± Ê ± o # " # r 1 r dr u du u 0 (1) È È ± ± o ± o ± o ± ± o # " " # \$Î# \$Î# ! " 3 (b) Use the same substitution for u as in part (a); r 1 u 0, r 1 u 0 o ± r 1 u du 0 È È ± o ± o # " # 3. (a) Let u tan x du sec x dx; x 0 u 0, x u 1 o o Ê o o Ê o # 1 4 tan x sec x dx u du 0 41 # # " ! " o ± o ’“ u1 (b) Use the same substitution as in part (a); x u 1, x 0 u 0 o ± ± 1 4 tan x sec x dx u du 0 # # ! ± " o ± o ± u 4. (a) Let u cos x du sin x dx du sin x dx; x 0 u 1, x u 1 o ± Ê ± o o Ê o o Ê o ± 1 3 cos x sin x dx 3u du u ( 1) (1) 2 1 \$ \$ \$ ± ± " " o ± o ± o ± ± ± ± o cd (b) Use the same substitution as in part (a); x 2 u 1, x 3 u 1 o o ± 3 cos x sin x dx 2 21 31 ± o ± o

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330 Chapter 5 Integration 5. (a) u 1 t du 4t dt du t dt; t 0 u 1, t 1 u 2 o ± Ê o Ê o o Ê o o Ê o %\$ \$ " 4 t 1 t dt u du '' 01 12 \$% \$ \$ " # " ab ’“ ± o oo ² o 41 6 1 6 1 6 1 6 u2 1 1 5 (b) Use the same substitution as in part (a); t 1 u 2, t 1 u 2 o ² Ê o o Ê o t 1 t dt 0 \$ \$ " ± o o 4 6. (a) Let u t 1 du 2t dt du t dt; t 0 u 1, t 7 u 8 o ± Ê o Ê o oÊo o Êo # " # È t t 1 dt u du u (8) (1) 78 ± ‘ˆ ˆ ˆ‰ˆ‰ #" Î \$ % Î \$ % Î \$ % Î \$ "Î\$ "" ## ) " ± o o o ² o 33 3 4 5 48 8 8 (b) Use the same substitution as in part (a); t 7 u 8, t 0 u 1 o ² È t t 1 dt u du u du ' 1 8 Î \$ " Î \$ "Î\$ ± o o ² o ² 45 8 7. (a) Let u 4 r du 2r dr du r dr; r 1 u 5, r 1 u 5 o ± Ê o o ² # " # dr 5 u du 0 ± ² " # ± # 15 5r 4r (b) Use the same substitution as in part (a); r 0 u 4, r 1 u 5 oÊo oÊo dr 5 u du 5 u 5 (5) 5 (4) 04 5r 8 ² " ± # ± " ± " ± " & % ² o ² ² ² o ± ˆ 8. (a) Let u 1 v du v dv du 10 v dv; v 0 u 1, v 1 u 2 o ± Ê o \$Î# "Î# # 32 0 3 È dv du u du ' 1 2 10 v 1v u3 3 3u 3 1 3 20 20 20 20 1 10 È ² " ± # # " # o ² o ² ² o ˆ‰ ± ±
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5.6 Area entre curvas - Section 5.6 Substitution and Area...

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