How to Compute Lengths of Arcs

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How to Compute Lengths of Arcs

• Approximating Arcs by Polygon Arcs
• Arc Length Formula
• Parametric Curves
• Inverse of sine as an Arc Length Function
• General Arc Length Functions

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Approximating Lengths of Arcs
We approximate the length of an arc on a graph of
a function f and corresponding to an interval
a,b by polygon arcs.
We obtain these polygon arcs by looking at a
division of the interval a,b into n
subintervals of length ?x(b-a)/n by points xk,
ax0ltx1ltx2ltltxn-1ltxnb. The approximating
polygon arc is formed by joining the points
xk-1,f(xk-1) to the points xk,f(xk) by
straight line segments.
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Approximating Lengths of Arcs
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The Circumference of a Circle
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Length of a Catenary Curve
The function cosh(x) has the interesting
property that the length of the red arc on the
graph of the function equals the area under the
red graph (and above the x-axis). There is one
other such function. Can you find it?
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Parametric Curves
Often it is not possible or practical to define a
curve in the plane by its equation. One may use
a parametric representation of the curve instead.
Such a representation is a mapping t
(x(t),y(t)) of an interval to the plane.
For example, a parametric representation of the
circle x2y21 with radius 1 is given by
x(t)cos(t), y(t)sin(t). This mapping maps the
t-axis onto the circle. The parameterization
covers the unit circle infinitely many times.
Restricting the parameter t to the interval
0,2p) one gets a parameterization of the circle
which parameterization covers the points of the
circle only once.
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Circumference of a Circle
Using the parametric representation of a circle
of radius r, the computation of the length of the
curve in question was simpler than the
computation based on the representation of upper
half of the circle as a graph of a function.
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Length of a Lemniscate
These computations are technical and can be best
done with Maple. Parametric representation of the
lemniscate is the best way to compute its length.
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Arc Length Functions
Assume that -1 ? t ? 1. Let f(t) be the
function defined, for positive values of t, by
saying that f(t) the length of the red arc on
the circle with radius 1. For negative values of
t define the function f as negative of the
corresponding arc length. By its definition the
function f has the property
-?/2? f(t) ? ?/2 for all t, -1 ? t ? 1.
By the definition of angles, f(t) is the angle,
in the above picture, with vertex at the origin
(since the circle has radius 1, this is how
angles are measured).
The length of the green side of the right angle
triangle in the figure is then sin(f(t)). By the
construction we now have
sin(f(t))t, i.e. f(t)arcsin(t) since -?/2?
f(t) ? ?/2. One concludes that the arcsin
function is an arc length function.
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General Arc Length Functions
A general arc length function is a function whose
values are determined by lengths of arcs on
graphs of other functions. For xgta, define the
function L by setting L(x) the length of
the red arc on the graph of a given
function f.
Problem Which functions are arc length functions
of other functions?
Previous considerations show that arcsin(x) is an
arc length function as defined above. Also
arccos(x) is such an arc length function.
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Derivatives of Arc Length Functions
Let L be an arc length function corresponding
to a function f, i.e. L(x) the length of the
red graph.
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Existence of Arc Length Functions