5
THE INTEGRAL
5.1 Approximating and Computing Area
Preliminary Questions
1.
Suppose that
[
2
,
5
]
is divided into six subintervals. What are the right and left endpoints of the subintervals?
SOLUTION
If the interval
[
2
,
5
]
is divided into six subintervals, the length of each subinterval is
5
−
2
6
=
1
2
. The right
endpoints of the subintervals are then
5
2
,
3
,
7
2
,
4
,
9
2
,
5, while the left endpoints are 2
,
5
2
,
3
,
7
2
,
4
,
9
2
.
2.
If
f
(
x
)
=
x
−
2
on
[
3
,
7
]
, which is larger:
R
2
or
L
2
?
On
[
3
,
7
]
, the function
f
(
x
)
=
x
−
2
is a decreasing function; hence, for any subinterval of
[
3
,
7
]
,the
function value at the left endpoint is larger than the function value at the right endpoint. Consequently,
L
2
must be larger
than
R
2
.
3.
Which of the following pairs of sums are
not
equal?
(a)
4
X
i
=
1
i
,
4
X
`
=
1
`
(b)
4
X
j
=
1
j
2
,
5
X
k
=
2
k
2
(c)
4
X
j
=
1
j
,
5
X
i
=
2
(
i
−
1
)
(d)
4
X
i
=
1
i
(
i
+
1
),
5
X
j
=
2
(
j
−
1
)
j
(a)
Only the name of the index variable has been changed, so these two sums
are
the same.
(b)
These two sums are
not
the same; the second squares the numbers two through Fve while the Frst squares the
numbers one through four.
(c)
These two sums
are
the same. Note that when
i
ranges from two through Fve, the expression
i
−
1 ranges from one
through four.
(d)
These two sums
are
the same. Both sums are 1
·
2
+
2
·
3
+
3
·
4
+
4
·
5.
4.
Explain why
100
∑
j
=
1
j
is equal to
100
∑
j
=
0
j
but
100
∑
j
=
1
1 is not equal to
100
∑
j
=
0
1.
The Frst term in the sum
∑
100
j
=
0
j
is equal to zero, so it may be dropped. More speciFcally,
100
X
j
=
0
j
=
0
+
100
X
j
=
1
j
=
100
X
j
=
1
j
.
On the other hand, the Frst term in
∑
100
j
=
0
1 is not zero, so this term cannot be dropped. In particular,
100
X
j
=
0
1
=
1
+
100
X
j
=
1
1
6=
100
X
j
=
1
1
.
5.
We divide the interval
[
1
,
5
]
into 16 subintervals.
(a)
What are the left endpoints of the Frst and last subintervals?
(b)
What are the right endpoints of the Frst two subintervals?
Note that each of the 16 subintervals has length
5
−
1
16
=
1
4
.
(a)
The left endpoint of the Frst subinterval is 1, and the left endpoint of the last subinterval is 5
−
1
4
=
19
4
.
(b)
The right endpoints of the Frst two subintervals are 1
+
1
4
=
5
4
and 1
+
2
³
1
4
´
=
3
2
.
6.
Are the following statements true or false?
(a)
The right-endpoint rectangles lie below the graph if
f
(
x
)
is increasing.
(b)
If
f
(
x
)
is monotonic, then the area under the graph lies between
R
N
and
L
N
.
(c)
If
f
(
x
)
is constant, then the right-endpoint rectangles all have the same height.
(a)
±alse. If
f
is increasing, then the right-endpoint rectangles lie above the graph.
(b)
True. If
f
(
x
)
is increasing, then the area under the graph is larger than
L
N
but smaller than
R
N
; on the other hand,
if
f
(
x
)
is decreasing, then the area under the graph is larger than
R
N
but smaller than
L
N
.
(c)
True. The height of the right-endpoint rectangles is given by the value of the function, which, for a constant function,
is always the same.