Graphics Reference
In-Depth Information
I
(
b
c
)
f
(
b
)
(
c
a
)
f
(
a
)
(
b
a
)(
f
(
b
)
f
(
a
)). Draw a
(v)
C
figurefor
C
I
.
(vi) (
C
I
)
C
I
0.
I
area
C
.
8 All the inequalities would have been reversed.
9
(i)
n
maximal rectangles with equal width inscribed in the area
bounded by
y
1/
x
,
x
b
, the
x
-axis and
x
ba
havearea
ba
b
1
1
b
)
b
)
...
n
b
(1/
n
)(
ba
b
(2/
n
)(
ba
1
b
(
n
/
n
)(
ba
b
)
a
1
1
1
1)
1)
...
(1/
n
)(
a
(2/
n
)(
a
n
1
1
1
1
(
n
/
n
)(
a
1)
which is thearea of
n
maximal rectangles with equal width
inscribed in the area bounded by
y
1/
x
,
x
1, the
x
-axis and
x
a
.
(ii) Since
y
1/
x
is monotonic decreasing for positive
x
, theareas
under the curve are well defined by qn 8. The two areas are each
equal to thelimit of thesums of theareas of theinscribed
rectangles as
n
.
(iii) Area under graph on [1,
ab
]
area under graph on [1,
b
]
area under graph on [
b
,
ab
]
area under graph on [1,
b
]
area under graph on [1,
a
] by (ii).
(iv) lim
n
(
lim
n
(
lim
n
(
(
ab
)
1)
a
1)
b
1) as
n
.
See also qn 4.41.
10 Let
f
(
x
)
: both functions aremonotonic for
positive
x
. On a positiveintrval
x
and
g
(
x
)
x
f
g
0, while
f
g
0 if all
integrals were positive.
11 Inscribed area
0. Circumscribed area
1. Not equal, so no integral.
12
A
f
(
x
)
A
,so
A
(
b
a
)
f
A
(
b
a
) and
f
A
(
b
a
).
13 The same definition is necessary when
A
B
.
14 Supposethat
A
C
(the argument can be easily adapted for other
orderings of these numbers).
B
A
(
c
h
a
)
2
hA
B
(
b
c
h
)
f
A
(
c
h
a
)
2
hC
B
(
b
c
h
)
