5128_Ch07_pp378-433

Section 7.4 Lengths of Curves 413
P
Slope sin' (c k )
x k
x k–1 c k x k
y k
Figure 7.34 The portion of the sine
curve above x k1 , x k . At some c k in the
interval, sin c k y k x k , the slope of
segment PQ. (Example 1)
d
c
0
y
(a, c)
a
√⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
(x k ) 2 (y k ) 2
y f(x)
x k – 1
x k
Q
x k
y k
(b, d)
Figure 7.35 The graph of f, approximated
by line segments.
P
Q
b
x
Length of a Smooth Curve
We are almost ready to define the length of a curve as a definite integral, using the procedure
of Example 1. We first call attention to two properties of the sine function that came
into play along the way.
We obviously used differentiability when we invoked the Mean Value Theorem to replace
Δy k Δx k by sinc k for some c k in the interval x k1 , x k . Less obviously, we used
the continuity of the derivative of sine in passing from 1 si nc k 2 Δx k to the
Riemann integral. The requirement for finding the length of a curve by this method, then, is
that the function have a continuous first derivative. We call this property smoothness. A
function with a continuous first derivative is smooth and its graph is a smooth curve.
Let us review the process, this time with a general smooth function f x. Suppose the
graph of f begins at the point a, c and ends at b, d, as shown in Figure 7.35. We partition
the interval a x b into subintervals so short that the arcs of the curve above them are
nearly straight. The length of the segment approximating the arc above the subinterval
x k1 , x k is Δx 2 k . Δy 2 k The sum Δx 2 k Δy 2 k approximates the length of the curve. We
apply the Mean Value Theorem to f on each subinterval to rewrite the sum as a Riemann sum,
x k
2 y k
y
x
2 1 (
)
2
k
k
x k
1 fc k 2 x k .
Passing to the limit as the norms of the subdivisions go to zero gives the length of the curve as
L b
a
1 fx 2 dx b
a
1 ( d d
We could as easily have transformed Δx 2 k Δy 2 k into a Riemann sum by dividing
and multiplying by Δy k , giving a formula that involves x as a function of y say, x gy
on the interval c, d:
L x k 2 y 2 2
k
y
y
k ( 1 xk
y
y
k
k
)
k
y
) x
2
dx.
1 gc 2 k y k .
For some point
c k in (x k 1 , x k )
For some c k
in (y k1 , y k )
The limit of these sums, as the norms of the subdivisions go to zero, gives another reasonable
way to calculate the curve’s length,
L d
1 gy 2 dy d
2
( 1 d x
d
) dy.
y
c
c
Putting these two formulas together, we have the following definition for the length of a
smooth curve.
DEFINITION
Arc Length: Length of a Smooth Curve
If a smooth curve begins at a, c and ends at b, d, a b, c d, then the length
(arc length) of the curve is
L b
L d
a
1 ( d d
c
1 ( d d
y
) x
2
dx
2
x
y ) dy
if y is a smooth function of x on a, b;
if x is a smooth function of y on c, d.