Solução_Calculo_Stewart_6e

F.
TX.10
SECTION 7.7 APPROXIMATE INTEGRATION ¤ 329
5. f(x) =x 2 sin x, ∆x = b − a
n = π − 0 = π 8 8
(a) M 8 = π 8 f π
16 + f 3π
16 + f 5π
16 + ···+ f 15π
16 ≈ 5.932957
(b) S 8 =
π
8 · 3 f(0) + 4f π
8 +2f 2π
8 +4f 3π
8 +2f 4π
8 +4f 5π
8 +2f 6π
8 +4f 7π
8 + f(π)
≈ 5.869247
Actual: π
0 x2 sin xdx = 84 −x 2 cos x π
+2 π
x cos xdx = 83 −π 2 (−1) − 0 +2 cos x + x sin x π
0 0 0
= π 2 +2[(−1+0)− (1 + 0)] = π 2 − 4 ≈ 5.869604
Errors: E M = actual − M 8 = π
0 x2 sin xdx− M 8 ≈−0.063353
E S = actual − S 8 = π
0 x2 sin xdx− S 8 ≈ 0.000357
7. f(x) = 4√ 1+x 2 , ∆x = 2 − 0 = 1 8 4
(a) T 8 = 1
4 · 2 f(0) + 2f 1
4 +2f 1
2 + ···+2f 3
2 +2f 7
4 + f(2) ≈ 2.413790
(b) M 8 = 1 4 f 1
8 + f 3
8 + ···+ f 13
8 + f 15
8 ≈ 2.411453
(c) S 8 = 1
4 · 3
f(0) + 4f
1
4
+2f
1
2
+4f
3
4
+2f(1) + 4f
5
4
+2f
3
2
+4f
7
4
+ f(2)
≈ 2.412232
9. f(x) = ln x
1+x , ∆x = 2 − 1 = 1 10 10
(a) T 10 = 1 [f(1) + 2f(1.1) + 2f(1.2) + ···+2f(1.8) + 2f(1.9) + f(2)] ≈ 0.146879
10 · 2
(b) M 10 = 1 [f(1.05) + f(1.15) + ···+ f(1.85) + f(1.95)] ≈ 0.147391
10
(c) S 10 = 1 [f(1) + 4f(1.1) + 2f(1.2) + 4f(1.3) + 2f(1.4) + 4f(1.5) + 2f(1.6) + 4f(1.7)
10 · 3
+2f(1.8) + 4f(1.9) + f(2)]
≈ 0.147219
1
11. f(t) =sin(e t/2 2
), ∆t =
− 0 = 1
8 16
(a) T 8 = 1
16 · 2 f(0) + 2f 1
16 +2f 2
16 + ···+2f 7
16 + f 1
2 ≈ 0.451948
(b) M 8 = 1
16 f 1
32 + f 3
32 + f 5
32 + ···+ f 13
32 + f 15
32 ≈ 0.451991
(c) S 8 = 1
16 · 3 f(0) + 4f 1
16 +2f 2
16 + ···+4f 7
16 + f 1
2 ≈ 0.451976
13. f(t) =e √t sin t, ∆t = 4 − 0 = 1 8 2
(a) T 8 = 1
2 · 2 f(0) + 2f 1
2 +2f(1) + 2f 3
2 +2f(2) + 2f 5
2 +2f(3) + 2f 7
2 + f(4) ≈ 4.513618
(b) M 8 = 1 2 f 1
4 + f 3
4 + f 5
4 + f 7
4 + f 9
4 + f 11
4 + f 13
4 + f 15
4 ≈ 4.748256
(c) S 8 = 1
2 · 3
f(0) + 4f
1
2
+2f(1) + 4f
3
2
+2f(2) + 4f
5
2
+2f(3) + 4f
7
2
+ f(4)
≈ 4.675111
15. f(x) = cos x
x , ∆x = 5 − 1 = 1 8 2
(a) T 8 = 1
2 · 2 f(1) + 2f 3
2 +2f(2) + ···+2f(4) + 2f 9
2 + f(5) ≈−0.495333
(b) M 8 = 1 2 f 5
4 + f 7
4 + f 9
4 + f 11
4 + f 13
4 + f 15
4 + f 17
4 + f 19
4 ≈−0.543321
(c) S 8 = 1
2 · 3
f(1) + 4f
3
2
+2f(2) + 4f
5
2
+2f(3) + 4f
7
2
+2f(4) + 4f
9
2
+ f(5)
≈−0.526123