Thomas Calculus 13th [Solutions]

352 Chapter 5 Integration
34. (a) (b) (c)
35. (a) (b) (c)
36. (a) (b) (c)
37. | x1 x0
| |1.2 0| 1.2, | x2 x1 | |1.5 1.2| 0.3, | x3 x2
| 2.3 1.5 0.8, | x4 x3
| 2.6 2.3 0.3,
and | x5 x4
| |3 2.6| 0.4; the largest is || P || 1.2.
38. | x1 x0| | 1.6 ( 2)| 0.4,| x2 x1
| | 0.5 ( 1.6) | 1.1,| x3 x2
| | 0 ( 0.5) | 0.5,
| x4 x3| |0.8 0| 0.8,and | x5 x 4| |1 0.8| 0.2; the largest is || P || 1.1.
39.
f ( x) 1
2
x
Let x 1 0 1
n n
and c i
i i x
n
. The right-hand sum is
n
n
2
2 n
1 c 1 1
i 1 i 1 2 2
n i
n n n 3
i 1
i 1
n i 1
3 n
n 1 2
i 1 n( n 1)(2n
1)
3 2
1 2n 3n n
3 3
3
3
n n i 1
6n
6n
3
2
1
n
2
1 n n . Thus, lim 1 c
2 1
6
i
n
n
i 1
2
3 1
n n2
lim 1 1 1 2
n
6 3 3
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