Thomas Calculus 13th [Solutions]

CHAPTER 5
INTEGRATION
5.1 AREA AND ESTIMATING WITH FINITE SUMS
1.
2
f ( x)
x Since f is increasing on [0, 1], we use left endpoints to
obtain lower sums and right endpoints to obtain upper
sums.
(a)
(b)
(c)
(d)
1 0
x
1 and
2 2
x
x
x
1 0 1
4 4
1 0 1
2 2
1 0 1
4 4
xi
i x
1
i a lower sum is i
2
1 1 2 1
2
0
1
2
2 2 2 2 8
i 0
and xi
i x
3
i a lower sum is i
2
1 1 2 2
0 1 1
2
3
2
1 7 7
4
4 4 4 4 2 4 4 8 32
i 0
2
and x i
i i x an upper sum is i
2 1 1 1
2 2
1 5
2
2 2 2 2 8
i 1
4
and x i
i i x an upper sum is i
2 1 1 1
2
1
2
3
2 2
1 1 30 15
4
4 4 4 4 2 4 4 16 32
i 1
2.
f ( x)
3
x Since f is increasing on [0, 1], we use left endpoints to
obtain lower sums and right endpoints to obtain upper
sums.
(a)
(b)
(c)
(d)
x
x
x
x
1 0 1
2 2
1 0 1
4 4
1 0 1
2 2
1 0 1
4 4
1
and x i
i i x a lower sum is i
3
1 1 3 1
3
0
1
2
2 2 2 2 16
i 0
3
and x i i x
i a lower sum is i
3 1 1 3 3
0 1 1
3
4
4 4 4 4 2
i 0
2
and x
i
i i x an upper sum is i
3 1 1 1
3 3
1 1 9 9
2
2 2 2 2
2 8 16
i 1
4
and x
i
i i x an upper sum is i
3 1
3 3 3
1 1 1
3
1
4
4 4 4 4 2
i 1
3
3
36 9
4 256 64
3 100 25
4 256 64
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