Answer Key
845
6.14 . 12 π units 3 6.15 . 11 π 6 units 3 6.16 . π 6 units 3 6.17 . Use the method of washers; V = ∫ −1 1 π ⎡ ⎣ ⎛
2 ⎤
2
⎞ ⎠
⎛ ⎝ x 2
⎞ ⎠
⎦ dx
⎝ 2− x 2
−
6.18 . 1 6 ⎠ ≈1.697 6.19 . Arc Length ≈ 3.8202 6.20 . Arc Length = 3.15018 6.21 . ⎛ ⎝ 5 5−1 ⎞ ⎛ ⎝ 5 5−3 3 ⎞ ⎠ ≈3.133
π 6
6.22 . 12 π 6.23 . 70/3 6.24 . 24 π 6.25 . 8 ft-lb 6.26 . Approximately 43,255.2 ft-lb 6.27 . 156,800 N 6.28 . Approximately 7,164,520,000 lb or 3,582,260 t 6.29 . M =24, x – = 2 5 m 6.30 . (−1, −1) m 6.31 . The centroid of the region is (3/2, 6/5). 6.32 . The centroid of the region is (1, 13/5). 6.33 . The centroid of the region is (0, 2/5). 6.34 . 6 π 2 units 3 6.35 . a. d dx ln ⎛ ⎝ 2 x 2 + x ⎞ ⎠ = 4 x +1 2 x 2 + x b. d dx ⎛ ⎝ ln ⎛ ⎝ x 3 ⎞ ⎠ ⎞ ⎠ 2 = 6 ln ⎛ ⎝ x 3 ⎞ ⎠ x
3 ln |
x 3 +6 | + C
2 x 3 +6
6.36 . ∫ x 6.37 . 4 ln2 6.38 .
dx = 1
⎛ ⎝ ⎜ e x
⎞ ⎠ ⎟ = e x
2 e 5 x
2 −5 x
d dx
(2 x −5)
a.
3
⎛ ⎝ e 2 t
⎞ ⎠
b. d dt
=6 e 6 t
6.39 . ∫ 4 e 3 x
e −3 x + C
dx = − 4 3
6.40 .
⎛ ⎝ 4 t 3
⎞ ⎠
t 4
4
a. d
=4 t
(ln4)
dt 4
⎛ ⎝ x 2 +1
⎞ ⎠ =
d dx log 3
x
⎛ ⎝ x 2 +1
⎞ ⎠
b.
(ln3)
6.41 . ∫ x 2 2 x 3
3
2 x
dx = 1 + C 6.42 . There are 81,377,396 bacteria in the population after 4 hours. The population reaches 100 million bacteria after 3 ln2
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