Graphics Reference
In-Depth Information
Likewise on [0, 1]
f
is strictly decreasing and again the polygonal arc
length is bounded above by 2. So all polygonal arc lengths, by qn 43,
arebounded aboveby 4.
46 Thefunction
g
is bounded on [
a
,
b
] and so integrable. Working on the
subdivision
1
a
x
x
x
...
x
b
1 and letting
m
inf
g
(
x
)
x
x
x
, and
M
sup
g
(
x
)
x
x
x
,
from qn 45,
m
(
x
x
)
polygonal arc length of
f
for this
subdivision
).
So sup (lower sums)
sup (polygonal arc lengths)
arc length.
So lower integral of
g
on [
a
,
b
]
arc length of
f
on [
a
,
b
].
M
(
x
x
47 The
z
-subdivision is thesameas the
x
-subdivision but with some
additional intervening points. The additional points increase the
polygonal arc length by qn 43.
The
z
-subdivision is thesameas the
y
-subdivision but with some
additional intervening points, so if
y
z
, then
y
z
for some
p
,
so [
z
,
z
]
[
y
,
y
] for
s
k
1, . . .,
k
p
.
So sup
g
(
x
)
z
x
z
sup
g
(
x
)
y
x
y
for
s
k
1, . . .,
k
p
.
Polygonal arc length on
x
-subdivision
polygonal arc length on
z
-subdivision
upper sum on
z
-subdivision (using qn 45)
upper sum on
y
-subdivision.
so every polygonal arc length
every upper sum.
So sup (polygonal arc lengths)
every upper sum.
So arc length is a lower bound to the upper sums.
So arc length
inf (upper sums)
upper integral.
48 Lower integral
upper integral (from qns 46 and 47). But
g
is bounded and continuous on [
a
,
b
], so
g
is integrable on [
a
,
b
], and
its upper integral and lower integral are equal. So arc length
arc length
integral.
49
(i)
A
is continuous by qn 10.50, thecontinuity of integrals.
(ii) If
t
s
A
(
t
)
1
1, then
A
(
s
)
0.
(iii)
A
is differentiable by the Fundamental theorem, qn 10.54.
50
(i) By qn 10.48, definition.
dx
(1
x
)
.
(ii)
A
(0)
Putting
u
x
,
dx
du
)
)
A
(0).
(1
x
(1
u
