Y 4x 4 x dx 4 x2 2 2 4x dx 16 x2 2 16 16 x2 12 12 37

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Unformatted text preview: x2 1 1, du 1 3 dx 2x dx. 1 2 1 x2 1 1 3 2x dx 12 x 8 1 1 4 1 0 67. Let u 2 x3 1, du 1 dx 3x2 dx 2 1 3 2 2x2 x3 1 x3 1 1 12 3x2 dx 2 x3 1 3 32 43 x 9 4 27 9 1 32 2 1 2 32 1 22 12 8 9 2 202 Chapter 4 Integration 69. Let u 4 0 2x 1 2x 1, du 1 dx 2 dx. 1 2 4 4 2x 0 1 12 2 dx 2x 1 0 9 1 2 71. Let u 9 1 1 1 x1 x, x 1, u x 1 x, du 1 dx. 2x 9 x 2 2 dx 2 1 1 x 2 1 2x dx 2 1 x 9 1 1 2 1 1 2 73. u 2 u, dx 1. When x 0 du 2, u 2 1 When x 2 0. 0 1 2 x dx u 1 u du 1 u3 2 u1 2 du 25 u 5 2 23 u 3 0 2 1 2 5 2 3 4 15 2 75. 0 cos 2 x dx 3 1, x 0, u 7 3 2 sin x 2 3 u 1, dx 1. When x 2 0 3 2 3 2 33 4 77. u x du 7, u 8 When x Area 8. 1 3 3 x3 x 0 1 dx 1 8 u u4 1 u du 3 u1 du 37 u 7 3 34 u 4 8 3 1 384 7 12 3 7 3 4 1209 28 79. A 0 2 sin x sin 2x dx 2 cos x 1 cos 2x 2 1 dx 2 7 4 0 2 3 81. Area 2 sec2 x dx 2 2 3 2 2 sec2 x 2 2 tan x 2 2 2 3 2 3 1 4 83. 0 x 2x 3 1 dx 3.333 10 3 85. 3 xx 15 3 dx 28.8 144 5 3 87. 0 5 cos 6 d 7.377 −1 −1 5 0 0 8 −1 −1 4 89. 2x 2x 1 2 dx 1 2 dx 1 2 2x 4x2 1 2 2 dx 4x 1 dx C1 1 2x 6 43 x 3 1 . 6 1 3 C1 x 43 x 3 C2 2x2 x 1 6 C1 2x2 They differ by a constant: C2 S ection 4.5 Integration by Substitution 203 91. f x 2 x2 x2 x2 x2 1 is even. 2 93. f x x2 dx 8 3 272 15 2 x5 5 x3 3 2 0 2 x x2 x x2 1 3 is odd. 0 1 dx 2 0 x4 32 5 1 3 dx 2 2 2 2 95. 0 x2 dx 0 x3 3 2 0 2 8 ; the function x2 is an even function. 3 x2 dx 0 2 (a) 2 2 x2 dx x2 dx 0 8 3 8 3 4 4 2 2 (b) 2 0 x2 dx 3x2 dx 2 2 0 x2 dx 2 16 3 8 (c) x2 dx 0 (d) 3 0 x2 dx 4 4 4 97. 4 x3 6x2 2x 3 dx 4 x3 2x dx 4 6x2 3 dx 0 2 0 6x2 3 dx 2 2x3 3x 0 232 99. Answers will vary. See “Guidelines for Making a Change of Variables” on page 292. 2 101. f x x x2 1 2 is odd. Hence, 2 x x2 1 2 dx 0. 103. dV dt Vt V0 V1 k t t k 1 k 2 1 k 1 C C 2 2 dt k t 1 C 500,000 400,000 200,000 and Solving this system yields k C 300,000. Thus, Vt When t b 200,000 t1 4, V 4 300,000. $340,000. 105. 1 b (a) (b) (c) a 74.50 a 43.75 sin 262.5 262.5 cos cos cos t 6 t 6 t dt 6 3 0 6 3 12 0 1 b a 74.50t 262.5 262.5 cos t 6 b a 1 74.50t 3 1 74.50t 3 1 74.50t 12 1 223.5 3 1 447 3 1 894 12 102.352 thousand units 223.5 262.5 102.352 thousand units 74.5 thousand units 262.5 262.5 262.5 t 6 204 Chapter 4 1 b (a) (b) a b Integration 1 b a 1 cos 60 t 30 1 60 107. 2 sin 60 t a cos 120 t dt 1 120 1 30 0 1 b sin 120 t a 1 1 60 1 1 240 0 0 1 cos 60 t 30 1 cos 60 t 30 1 sin 120 t 120 1 sin 120 t 120 60 0 1 240 1 30 1 120 1.382 amps 1 30 4 1.273 amps 1 30 240 0 30 2 22 2 1 1 30 1 cos 60 t 30 1 sin 120 t 120 1 30 5 1 30 (c) 0 30 0 0 amps 109. False 2x 111. True 10 10 10 10 1 2 dx 1 2 2x 1 2 2 dx 1 2x 6 1 3 C ax3 10 bx2 cx d dx 10 ax3 cx dx 10 bx2 d dx Even 0 2 0 bx2 d dx Odd 113. True 4 sin x cos x dx 2 sin 2x dx cos 2x C 115. Let u b x fx a h, then du b dx. When x h b a, u h a h. When x b, u b h. Thus, h dx a h f u du a h f x dx. Section 4.6 2 Numerical Integration x2 dx 0 2 1. Exact: Trapezoidal: 0 2 13 x 3 1 0 4 1 0 6 2 0 8 3 2 4 1 2 1 2 2 2.6667 21 2 2 x2 dx x2 dx 0 2 4 3 2 3 2 2 2 2 2 11 4 8 3 2.7500 2.6667 Simpson’s: 21 2 2 2 2 3. Exact: 0 2 x3 dx x3 dx 0 2 x4 4 1 0 4 1 0 6 2 4.000 0 Trapezoidal: Simpson’s: 0 2 4 1 2 1 2 3 21 3 3 2 4 3 2 3 2 3 2 3 3 17 4 24 6 4.2500 4.0000 x3 dx 21 3 2 3 S ection 4.6 Numerical Integration 205 2 5. Exact: 0 2 x3 dx x3 dx 0 2 14 x 4 1 0 8 1 0 12 2 4.0000 0 Trapezoidal: Simpson’s: 0 2 4 1 4 1 4 3 2 3 2 4 2 2 4 3 2 3 3 4 4 3 4 3 21 3 3 2 3 5 4 4 5 4 3 2 3 6 4 2 6 4 3 2 3 7 4 4 7 4 3 8 3 4.0625 8 4.0000 x3 dx 21 9 7. Exact: 4 9 x dx x dx 4 23 x 3 5 2 16 9 2 4 18 2 37 8 16 3 2 38 3 21 4 12.6667 2 47 8 2 26 4 2 57 8 2 31 4 2 67 8 3 Trapezoidal: 12.6640 9 Simpson’s: 4 x dx 5 2 24...
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