Graphics Reference
In-Depth Information
k
Â
b
lF
=
lim
P of a,b ,
pp
,
i
-
1
i
[
]
partition
P
Æ
0
i
=
1
provided that this limit exists. (|P| denotes the norm of the partition.) A parametric
curve that has a length is called a
rectifiable curve
.
Computing limits would be complicated. Fortunately, we can compute integrals
instead.
If F is C
1
, then l
b
F exists and
9.2.1. Proposition.
b
=¢
b
Ú
lF
F
.
a
Proof.
We shall only sketch a proof. For simplicity assume that n = 2. Let F(t) =
(x(t),y(t)) and
p
i
= (x
i
,y
i
) = F(t
i
). The Proposition basically follows from the mean value
theorem, which implies that
-=¢
()
-
(
)
xx
x tt
a
b
,
,
i
i
-
1
i
i
-
1
-=¢
()
(
)
yy
y tt
-
i
i
-
1
i
i
-
1
for some a, bŒ[t
i-1
,t
i
]. In other words,
2
2
(
)
(
)
pp
i
=-
xx
+-
yy
-
1
i
i
i
-
1
i
i
-
1
[
]
2
2
2
(
)
¢
()
+
()
=-
tt
x
a
y
b
.
i
i
-
1
Summing these expressions gives something very much like a Riemann sum that con-
verges to the desired integral.
9.2.2. Example.
To compute the lengths of
()
=
()
Œ
[]
Ft
tt
,,
t
01
01
02
,,
()
=
(
)
22
Œ
[]
G t
t
,
t
,
t
,
,
and
()
=
(
)
Œ
[
]
Ht
sin
t
,
sin
t
,
t
,
p
.
Solution.
By Proposition 9.2.1
1
1
Ú
Ú
lF
=
22
=
,
0
0
1
1
l G
=
22
t
=
2
,
and
0
p
2
p
2
Ú
lH
=
2
cos
t
=
2
.
0
0
We should note that although we can now compute lengths using integration, it
is still not that easy. The fact that there is a square root under the integral sign means
that it is in general pretty much impossible to compute the integral in closed form.
However, one can get good approximations using numerical analysis. Another obser-
