Graphics Reference
In-Depth Information
function
f
of qn 39 is greater than a lower sum for the function
g
on theintrval [
a
,
b
], and less than an upper sum for this function
provided
1. Useqn 10.39.
Deduce that the lower integral
1
a
b
g
arc length on [
a
,
b
].
47 Let
1
a
x
x
x
...
x
b
1 and
a
y
b
betwo subdivisions of theintrval
[
a
,
b
], and let the union of these two subdivisions be
a
z
y
y
...
y
z
z
...
z
b
.
Useqn 43 to show that
thepolygonal arc length of
f
with the'
x
' subdivision
thepolygonal arc length of
f
with the'
z
' subdivision.
Explain why
theuppr sum for
g
on the'
y
' subdivision
theuppr sum for
g
on the'
z
' subdivision.
Deduce from qn 46 that
any polygonal arc length of
f
on [
a
,
b
]
any upper sum for
g
on [
a
,
b
].
Deduce that the arc length on [
a
,
b
]
the upper integral
g
.
48 From qns 46 and 47 and the definition of the Riemann integral we
know that thearc length of
f
on [
a
,
b
] is equal to
g
.
Use qn 10.64 and 10.66 to show that the integral exists as an
improper integral even when
a
1or
b
1 or both.
Arc cosine
Wenow
define
an anglefunction (or arc length function)
A
:[
1, 1]
R by
dx
A
(
y
)
)
.
x
(1
49 Say why thefunction
A
is
(i) continuous,
(ii) monotonic decreasing,
1
(1
y
)
.
(iii)
differentiable on (
1, 1) with
A
(
y
)
50 Say why
(i)
A
(1)
0,
(ii)
A
(
1)
2
A
(0).
(iii)
A
is a bijection [
1, 1]
[0,
A
(
1)].
