Consider how to define the length of the graph of a function y = f
(x)
. Consider the
following picture.
PICT
which depicts a small right triangle attached as shown to the graph of a function
y = f
(x)
for x ∈
[a,b]
. If the triangle is small enough, this shows the length of the
curve joined by the hypotenuse of the right triangle is essentially equal to the
length of the hypotenuse. Thus,
. Thus the length of the curve is given by the definite integral,
∫ b∘ ---------
1 + f′(x)2dx
a
Alternatively, you could consider a partition of
[a,b]
,a = x_{0}< x_{1}<
⋅⋅⋅
< x_{n} = b and
observe that the length of the curve between the two points
(xi,f (xi))
and
(xi−1,f (xi−1))
is approximately the length of the line joining these two points and that
the length of the curve would be close to the sum of these as suggested in the following
picture.
PICT
Thus the length of the curve would be approximately the sum of the lengths of the
little straight lines in the above picture and this equals
dx. One would imagine that this
approximation would in the limit, as the norm of the partition converges to 0, have what
should be defined as the length of the curve as the limit.
This definition gives the right answer for the length of a straight line. To see this,
consider a straight line through the points
(a,b)
and
(c,d)
where a < c. Then the
right answer is given by the Pythagorean theorem or distance formula and is
∘ ----------------
(a− c)2 + (d− b)2
. What is obtained from the above initial value problem? The
equation of the line is f
(x)
= b +
(d−b)
c−a
(x− a)
and so f^{′}
(x)
=
(d−b)
c−a
. Therefore, by
the new procedure, the length is
∘ ------------ ∘ ------------
∫ c ( d− b)2 ( d− b)2 ∘ ----------------
1+ ----- dx = (c− a) 1+ ----- = (a− c)2 + (d− b)2
a c− a c− a
as hoped. Thus the new procedure gives the right answer in the familiar cases but it
also can be used to find lengths for more general curves than straight lines.
Summarizing,
Procedure 8.1.1To find the length of the graph of the function y = f
(x)
forx ∈
[a,b]
, compute
∫ ---------
b∘ ′ 2
a 1+ f (x) dx.
Here is another familiar example.
Example 8.1.2Find the length of the part of the circle having radius r which isbetween the points