Solução_Calculo_Stewart_6e

F.
358 ¤ CHAPTER 8 FURTHER APPLICATIONS OF INTEGRATION
TX.10
29. y =lnx ⇒ dy/dx =1/x ⇒ 1+(dy/dx) 2 =1+1/x 2 =(x 2 +1)/x 2 ⇒
2
x 2
+1
2
√ 1+x
2 1+x2 L =
dx =
dx 23 = − ln 1 x 2 1 x
1+√ 1+x 2
x
= √ √ 1+ 5
5 − ln
− √ 2+ln 1+ √ 2
2
31. y 2/3 =1− x 2/3 ⇒ y =(1− x 2/3 ) 3/2 ⇒
dy
dx = 3 (1 − 2 x2/3 ) 1/2 − 2 3 x−1/3 = −x −1/3 (1 − x 2/3 ) 1/2 ⇒
2 dy
= x −2/3 (1 − x 2/3 )=x −2/3 − 1. Thus
dx
L =4
1
0 1+(x
−2/3
− 1) dx =4 1
0 x−1/3 dx = 4 lim
1
3
t→0 + 2 x2/3 t
33. y =2x 3/2 ⇒ y 0 =3x 1/2 ⇒ 1+(y 0 ) 2 =1+9x. The arc length function with starting point P 0 (1, 2) is
s(x) = x √
x
1 1+9tdt=
2
(1 + 27 9t)3/2 = 2 (1 + 9x) 3/2 − 10 √
10 .
27
1
=6.
2
1
35. y =sin −1 x + √ 1 − x 2 ⇒ y 0 =
1
√ − x
√ = √ 1 − x
1 − x
2 1 − x
2 1 − x
2
⇒
1+(y 0 ) 2 (1 − x)2
=1+ = 1 − x2 +1− 2x + x 2
= 2 − 2x
1 − x 2 1 − x 2 1 − x = 2(1 − x)
2 (1 + x)(1 − x) = 2
1+x ⇒
2
1+(y0 ) 2 = . Thus, the arc length function with starting point (0, 1) is given by
1+x
x
x
s(x) = 1+[f 0 2
(t)] 2 dt =
1+t dt = √ 2 2 √ 1+t x
=2√ 2 √ 1+x − 1 .
0
0
0
37. The prey hits the ground when y =0 ⇔ 180 − 1 45 x2 =0 ⇔ x 2 =45· 180 ⇒ x = √ 8100 = 90,
since x must be positive. y 0 = − 2 x ⇒ 45 1+(y0 ) 2 =1+ 4 x 2 , so the distance traveled by the prey is
45 2
90
L = 1+ 4 4
45 2 x2 dx =
1+u
2 45
du u = 2
45 x,
2 du = 2
0
21
= 45 2
0
45 dx
1 u √ 1+u
2 2 + 1 ln u + √ 1+u 2 2 4
= √ 45
0 2 2 17 +
1
ln 4+ √ 17 =45 √ 17 + 45 ln 4+ √ 17 ≈ 209.1 m
2 4
39. Thesinewavehasamplitude1 and period 14, since it goes through two periods in a distance of 28 in., so its equation is
y =1sin 2π
14 x =sin π
7 x . The width w of the flat metal sheet needed to make the panel is the arc length of the sine curve
from x =0to x =28.Wesetuptheintegraltoevaluatew using the arc length formula with dy
dx = π 7 cos π
L = 28
0
1+ π
7 cos π
7 x2 dx =2 14
0
so we use a CAS, and find that L ≈ 29.36 inches.
x :
7
1+ π
cos π
7 7 x2 dx. This integral would be very difficult to evaluate exactly,
41. y = x
√
t3 − 1 dt ⇒ dy/dx = √ x
1
3 − 1 [by FTC1] ⇒ 1+(dy/dx) 2 =1+ √ x 3 − 1 2
= x
3
L = 4
√
x3 dx = 4
1
1 x3/2 dx = 2 5
x 5/2 4
1
= 2 5 (32 − 1) = 62 5 =12.4
⇒