30.
Definition of Arc Length
If the function given by
)
(
x
f
y
=
represents a smooth curve on the interval
[
]
b
a
,
, then the arc
length of
f
between
a
and
b
is given by
[
]
dx
b
a
x
f
s
2
)
(
1
∫
′
+
=
.
31.
Work and Hooke’s Law
1.
If an object is moved a distance
D
in the direction of an applied constant
force
F
, then the
work
W
done by the force is defined as
W = FD
.
2.
If an object is moved along a straight line by a continuously
varying
force
)
(
x
F
then the
work
W
done by the force as the object is moved from
to
b
x
a
x
=
=
is given by
W
=
∫
b
a
dx
x
F
)
(
.
3.
Hooke’s law says that the amount of force
F
it takes to stretch or compress a spring
x
units
from its natural length is proportional to
x
.
That is,
F
=
kx
, where
k
is the spring constant
measured in force units per unit length.
32.
Improper Integral
∫
b
a
dx
x
f
)
(
is an improper integral if
1.
f
becomes infinite at one or more points of the interval of integration, or
2.
one or both of the limits of integration is infinite, or
3.
both (1) and (2) hold.
33.
Parametric Form of the Derivative
If a smooth curve
C
is given by the parametric equations
)
(
and
)
(
t
g
y
x
f
x
=
=
, then the
slope of the curve
C
at
0
,
is
)
,
(
≠
÷
=
dt
dx
dt
dx
dt
dy
dx
dy
y
x
.
Note
:
The second derivative,
dt
dx
dx
dy
dt
d
dx
dy
dx
d
dx
y
d
÷
=
=
2
2
.
34.
Arc Length in Parametric Form
If a smooth curve
C
is given by
)
(
and
)
(
t
g
y
t
f
x
=
=
and these functions have continuous
first derivatives with respect to
t
for
b
t
a
≤
≤
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 Spring '08
 All
 Calculus, Arc Length, Polar Coordinates, Mathematical Series, Mathematical analysis, series A

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