Solução_Calculo_Stewart_6e

F.
360 ¤ CHAPTER 8 FURTHER APPLICATIONS OF INTEGRATION
TX.10
15. x = a 2 − y 2 ⇒ dx/dy = 1 2 (a2 − y 2 ) −1/2 (−2y) =−y/ a 2 − y 2 ⇒
1+
S =
2 dx
=1+
dy
a/2
0
y2
a 2 − y = a2 − y 2
2 a 2 − y + 2
y2
a 2 − y 2 =
2π
a
a/2
a 2 − y 2
a2 − y dy =2π 2
0
a2
a 2 − y 2
⇒
ady =2πa y a/2
0
a
=2πa
2 − 0 = πa 2 .
Note that this is 1 the surface area of a sphere of radius a, and the length of the interval y =0to y = a/2 is 1 the length of the
4 4
interval y = −a to y = a.
17. y =lnx ⇒ dy/dx =1/x ⇒ 1+(dy/dx) 2 =1+1/x 2 ⇒ S = 3
1 2π ln x 1+1/x 2 dx.
Let f(x) =lnx 1+1/x 2 .Sincen =10, ∆x = 3 − 1
10
= 1 5 . Then
S ≈ S 10 =2π · 1/5
3
[f(1) + 4f(1.2) + 2f(1.4) + ···+2f(2.6) + 4f(2.8) + f(3)] ≈ 9.023754.
The value of the integral produced by a calculator is 9.024262 (to six decimal places).
19. y =secx ⇒ dy/dx =secx tan x ⇒ 1+(dy/dx) 2 =1+sec 2 x tan 2 x ⇒
S = π/3
2π sec x √ 1+sec
0 2 x tan 2 xdx.Letf(x) =secx √ 1+sec 2 x tan 2 x.Sincen =10, ∆x = π/3 − 0
10
Then S ≈ S 10 =2π · π/30 π
f(0) + 4f +2f
3
30
2π
+ ···+2f
30
8π
+4f
30
The value of the integral produced by a calculator is 13.516987 (to six decimal places).
9π
π
+ f
30 3
≈ 13.527296.
21. y =1/x ⇒ ds = 1+(dy/dx) 2 dx = 1+(−1/x 2 ) 2 dx = 1+1/x 4 dx ⇒
2
S = 2π · 1
1+ 1 2
√
1 x x dx =2π x4 +1
4
√
u2 +1 dx =2π
1 4 1 x 3 1 u du 2 [u = x 2 , du =2xdx]
2
4
√ √
1+u
2
= π
du 24 1+u
2
= π − +ln
u + 4
1+u
1 u 2
u
2 1
= π − √ 17
+ln 4+ √ 17 + √ 2
− ln 1+ √ 2 = π √ √ √ √
4 1 4ln 17 + 4 − 4ln 2+1 − 17 + 4 2
4
23. y = x 3 and 0 ≤ y ≤ 1 ⇒ y 0 =3x 2 and 0 ≤ x ≤ 1.
S = 1
2πx 1+(3x
0 2 ) 2 dx =2π
3
√
0 1+u
2 1
du 6
u =3x 2 ,
du =6xdx
= π 3
3
0
√
1+u2 du
= π 30 .
21
= [or use CAS] π 1 u √ 1+u
3 2 2 + 1 ln u + √ 1+u 2 2 3
= π 3
√
0 3 2 10 +
1
ln 3+ √ 10 √ √
= π 2 6 3 10 + ln 3+ 10
25. S =2π
∞
1
1
y
1+
2 dy
∞
1
dx =2π 1+ 1 ∞
√
dx
1 x x dx =2π x4 +1
dx. Ratherthantryingto
4 1 x 3
evaluate this integral, note that √ x 4 +1> √ x 4 = x 2 for x>0. Thus, if the area is finite,
∞
√
x4 +1
∞
x 2 ∞
S =2π
dx > 2π
x 3
x dx =2π 1
dx. But we know that this integral diverges, so the area S is
3 x
infinite.
1
1