Math 20C Multivariable Calculus
Lecture 7
1
Slide 1
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$
%
The arc length of a curve in space
•
Arc length of a curve.
•
Arc length function.
•
Examples.
Slide 2
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The arc length of a curve is a number that
measures the extension of the curve
DeFnition 1
Let
r
(
t
)
be a continuously diferentiable
vectorvalued Function. The length oF the curve associated
with
r
(
t
)
For
t
∈
[
a, b
]
is the number given by
‘
ba
=
Z
b
a

r
0
(
t
)

dt.
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View Full DocumentMath 20C Multivariable Calculus
Lecture 7
2
Slide 3
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%
The arc length of a curve in space has the
following form in components
r
(
t
)
=
h
x
(
t
)
, y
(
t
)
, z
(
t
)
i
,
r
0
(
t
)
=
h
x
0
(
t
)
, y
0
(
t
)
, z
0
(
t
)
i
,
‘
ba
=
Z
b
a
q
[
x
0
(
t
)]
2
+ [
y
0
(
t
)]
2
+ [
z
0
(
t
)]
2
dt.
Suppose that the curve represents the path traveled by a particle in
space. Then, the length of the curve is the integral of the speed,

v
(
t
)

.
So in this case the length of the curve is the distance traveled by the
particle.
Slide 4
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 Spring '08
 Helton
 Calculus, Arc Length, Derivative, Multivariable Calculus, Velocity, arc length function

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