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# 5.6-5practice - 370 Chapter 5 Integration 94. A oe A" b A#...

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370 Chapter 5 Integration 94. A A A œ± "# Limits of integration: y x and y x x x œœ Ê œ \$& \$ & x x 0 x (x 1)(x 1) 0 a 1, b 0 Ê ²œÊ ² ±œÊ œ ² œ &\$ \$ "" and a 0, b 1; f (x) g (x) x x and ##"" ²œ ² f (x) g (x) x by symmetry about the origin, ## ² Ê A A 2A A 2 x x dx 2 # " ! ±œ Êœ ² œ ² ' 0 1 ab ’“ xx 46 %' 2 œ² œ ˆ‰ "" " 46 6 95. A A A Limits of integration: y x and y x , x 0 Ê œ Á x 1 x 1 , f (x) g (x) x 0 x Ê œ Ê œ ² œ²œ \$ A x dx ; f (x) g (x) 0 œ œ ² œ ² # " ! # ' 0 1 ’“ x 2x # # x A x dx 1 ; œÊ œ œ œ ² ± œ ±# ±# # ±" " " # " ' 1 2 ±‘ x AA A 1 œ ± œ±œ 96. Limits of integration: sin x cos x x a 0 œ Ê œ 1 4 and b ; f(x) g(x) cos x sin x œ ² 1 4 A (cos x sin x) dx [sin x cos x] ² œ ± ' 0 4 1 Î 1 Î% ! (0 1) 2 1 œ±² ± œ ² Š‹ È ÈÈ 22 97. (ln 2x ln x) dx ( ln x ln 2 ln x) dx (ln 2) dx (ln 2)(5 1) ln 2 ln 16 '' ' 11 1 55 5 ² ± œ œ % 98. A tan x dx tan x dx dx dx ln cos x ln cos x ± œ ² œ ² ' ' Î Î ÎÎ 40 4 0 03 0 3 ±± ! ±Î % Î\$ ! sin x sin x cos x cos x cd kk 1 1 ln 1 ln ln ln 1 ln 2 ln 2 ln 2 ²²œ ± œ È È 2 3 99. e e dx e e e 3 1 2 2 ' 0 ln 3 ab Š Š 2x x x ln 3 eee 9 8 ln 3 0 ² ² ² œ ² ² ² œ ² œ 2ln3 # # # # ! " ! 100. e e dx 2e 2e 2e 2e 2e 2e (4 1) (2 2) 5 4 1 ' 0 2 ln 2 ± ˆ x2 l n2 l 2ln2 0 Î± Î Î ± Î ± ! ! ² œ ± œ ± ² ± œ±²±œ²œ 101. A dx 2 dx; u 1 du 2x dx; x 0 u 1, x 2 u 5 œ ± Ê œ œ Ê œ œ Ê œ 20 2x 2x 1x ²² # A 2 du 2 ln u 2(ln 5 ln 1) 2 ln 5 Äœ œ œ ² œ ' 1 5 " & " u 102. A 2 dx 2 dx 2 2 ² ² œ ² ² œ # # # # " ±" x Ð Ñ ˆ ‰ ˆ ‰ˆ ‰ –— " # " # x ln 3 3 ln ln ln

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Section 5.6 Substitution and Area Between Curves 371 103. (a) The coordinates of the points of intersection of the line and parabola are c x x c and y c œÊ œ œ # È (b) f(y) g(y) y y 2 y the area of ±œ± ±œ Ê ÈÈÈ ˆ‰ the lower section is, A [f(y) g(y)] dy L 0 c œ± ' 2 y dy 2 y c . The area of œœ œ ' 0 c c È ±‘ 24 33 \$Î# \$Î# ! the entire shaded region can be found by setting c 4: A 4 . Since we want c to divide the œ œ 44 8 3 2 3 \$Î# region into subsections of equal area we have A 2A 2 c c 4 œ Ê œ L 32 4 \$Î# #Î\$ (c) f(x) g(x) c A [f(x) g(x)] dx c x dx cx 2 c ± Ê œ ± œ ± œ ± œ ± # # \$Î# L cc c c '' ÈÈ È ab’“ xc \$ \$Î# È c . Again, the area of the whole shaded region can be found by setting c 4 A . From the Ê œ 4 32 3 3 \$Î# condition A 2A , we get c c 4 as in part (b). Ê œ L 43 2 \$Î# #Î\$ 104. (a) Limits of integration: y 3 x and y 1 # 3 x 1 x 4 a 2 and b 2; Ê±œ ±Ê œÊœ ± œ ## f(x) g(x) 3 x ( 1) 4 x ± ± ± œ ± ab A 4 4x Êœ ± œ ± ' 1 2 ’“ # # ±# x 3 \$ 88 1 6 œ±± ±²œ±œ ˆ 8 8 16 32 3 3 (b) Limits of integration: let x 0 in y 3 x ± # y 3; f(y) g(y) 3 y 3 y ± œ ±±± ± 2(3 y) "Î# A 2 (3 y) dy 2 (3 y) ( 1) dy ( 2) 0 (3 1) ± œ ± ± ± œ ± œ ± ± ² 11 "Î# "Î# \$Î# ± \$ ±" ± 2(3 y) 4 \$ Î# (8) 2 105. Limits of integration: y 1 x and y œ² œ È 2 x È 1 x , x 0 x x 2 x (2 x) Ê² œ ÁÊ ²œÊœ± 2 x È # x 4 4x x 5x 4 0 Êœ±² Ê ±²œ (x 4)(x 1) 0 x 1, 4 (but x 4 does not Ê± ± œ Ê œ œ satisfy the equation); y and y Ê œ 2x 2 x xx 8 x x 64 x 4.
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## This note was uploaded on 04/17/2008 for the course MA 113 taught by Professor Massman during the Spring '08 term at Rose-Hulman.

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5.6-5practice - 370 Chapter 5 Integration 94. A oe A" b A#...

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