1. Left endpoint A
Above all, the Pythagorean theorem is a distance formula between two points.C
represents the straight-line distance between the two points, where the difference
between the x-coordinates of the points is ±a and the difference between the y-
coordinates of the points is ±b, or vice-versa, since any two points can be
represented in one x-y plane. The Pythagorean theorem is of fundamental
importance and is far-reaching in mathematics. Here is one example why:
Right endpoint B
Let’s say that the graph on the left
represents the path of an object
through two-dimensional space,
and that its x and y values at each
point represent perpendicular
components of displacement from a
reference point.To be clear, this
would actually be a parametric
function of some other variable
such as time, but right now that fact
is irrelevant. The left endpoint on
the graph of the object’s path
indicates the object’s location
relative to the reference point at
the beginning of the interval and
the right endpoint indicates its
relative location at the end of the
interval, which can be thought of as a time interval but not necessarily. Obviously,
the Pythagorean theorem allows us to calculate the shortest distance between the
object’s starting location and ending location, but it also allows us to calculate the
length of the curve on the graph, and since x and y are components of a parametric
function of some other variable and represent perpendicular directions in two-
dimensional space, the length of the curve (in the same units of measurement as
stated by the graph) is equal to the total length of the object’s path during the
interval, or more simply put, the total distance the object travelled during the
interval.
Unfortunately for us, we do not have formulas for lengths of nonlinear curves yet,
but we will, after we use the Pythagorean theorem, which only applies to line
segments, to help us derive such formulas. We can treat the straight-line distance
between endpoints as a crude approximation of the curve’s length, and we calculate
this distance using the Pythagorean theorem. What happens if we do this for two
subintervals and add up the distances between the endpoints of each to
approximate the curve’s length? The approximation will automatically be more
accurate, and in fact, as n, the number of subintervals taken into account, increases,
the accuracy of the approximations using n subintervals increases or stays the same.

2. More important than the number of subintervals is the requirement that all of their
lengths get smaller and smaller and approach zero. This infinitely small interval
involves infinitely small changes in x and y, called differentials. What matters more
than their respective quantities, which they do not actually have since they are
infinitely small, is their relative quantities, the ratio of the y differential to x
differential or vice versa, over each infinitely small subinterval.
The picture on the left shows Δx as h and Δy as
f(x+h) – f(x) over the interval from x to x+h. The
straight-line distance between the interval’s two
endpoints is ((Δx)2 + (Δy)2)1/2, the square root
of c2 in the Pythagorean equation. As h
approaches 0, the error in approximating the
curve length over each interval with the
Pythagorean straight-line distance formula
approaches 0, or in other words the Pythagorean
approximation of curve length becomes more
accurate as the interval over which it is used gets
smaller. If ((Δx)2 + (Δy)2)1/2 equals the curve
length over infinitely small intervals, how can
we add up the curve lengths of all the infinitely small subintervals to get the total
curve length of the whole interval? It is impossible to calculate the hypotenuse of
each right triangle formed by the x and y differentials over each subinterval because
differentials don’thave absolute sizes. However, as stated earlier, the x and y
differentials over infinitely small intervals do have relative sizes and that should
make sense because every point on a differentiable curve has a derivative that
represents the quotient dy/dx at that point, and dy/dx represents the Δy per
infinitely small Δx, where Δx and Δy are differentials.
((Δx)2 + (Δy)2)1/2 equals the curve length over an infinitely
small interval, where Δx and Δy are called differentials.
Δy = Δy Δx = dy dx because infinitely small Δy = dy and
---------- ----------
Δx dx infinitely small Δx = dx
Let y’ denote the derivative of y with respect to x even if the
two variables are actually independent of each other (due to
the fact that x and y model positions in perpendicular
directions of two dimensional space as part of a parametric
function), since all we care about right now is length.

3. (Δx)2 = (dx)2 (Δy)2 = (y’dx)2 = (y’)2(dx)2
((Δx)2 + (Δy)2)1/2 = ((dx)2 + (y’)2(dx)2)1/2 = ((dx)2(1 + (y’)2))1/2
= dx(1 + (y’)2)1/2 = length of curve over infinitely small interval
How do we add up all of the infinitely small curve lengths?
By integrating.
(1 (y')
2
)
1 / 2
dx
a
b
where a is the left endpoint and b is the right endpoint of the
interval over which we integrate.
For what we are trying to accomplish, which is derive a
formula for curve length, we can arbitrarily call y “f(x)” if the
curves we are dealing with represent y as functions of x,
because even though the example situation called for a
parametric functionwith x and y as separate functions of the
same variable, like t for time, all we may want to know is the
length of the curve on the graph and not what the length of the
curve represents or means in the specific situation in which it
is calculated. If y cannot be written as a function of x but x can
be written as a function of y, then the length of the curve over
the interval from y = c to y = d equals
(1 ( f '(y))
2
c
d
)
1 / 2
dy ,

4. or, using more common notation,
1 ( f '(y))
2
c
d
dy
*Picture taken from http://www.eldamar.org.uk/maths/calculus/node2.html