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8.1 Arc Length
Definition:
If the function given by
represents a smooth
curve on the interval [
a
,
b
], then the
arc length
of
f
between
a
and
b
is given by
Similarly, for a smooth curve given by
, the
arc
length
of
g
between
c
and
d
is given by
)
(
x
f
y
=
[ ]
dx
dx
dy
dx
x
f
L
b
a
b
a
∫
∫
+
=
+
=
2
2
1
)
(
'
1
[ ]
dy
dy
dx
dy
y
g
L
d
c
d
c
∫
∫
+
=
+
=
2
2
1
)
(
'
1
)
(
y
g
x
=
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View Full DocumentExamples:
1)
Find the arc length of the graph of
from
point
A
(0,1) to point
B
(8,5).
Solutions:
(
29
2
3
1
x
y
=

( 29
( 29
( 29
( 29
( 29
( 29
( 29
1
10
10
27
8
4
40
27
1
5
9
3
2
9
1
2
1
5
9
2
1
4
5
9
1
4
5
9
1
4
9
1
1
1
2
3
1
2
/
3
3/2
5
1
2
/
3
2
/
1
5
1
5
1
2
2
2
/
1
2
/
3

=

=

⋅
⋅
=

=

=
+
=
∴

=

+
=
+
⇒

=
⇒

±
=
∫
∫
∫
y
dy
y
dy
y
dy
dy
dx
L
y
y
dy
dx
y
dy
dx
y
x
d
c
Choosing the positive value of
x
, we have
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 Spring '08
 varies
 Calculus, Arc Length

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