Arc length and speed
Evans M. Harrell II
School of Mathematics
Georgia Tech
harrell@math.gatech.edu
Recall that there are two useful ways to represent a curve in the plane.
Many curves are graphs of functions, for example the semicircle:
> plot(sqrt(1-x^2),x=-1..1, y = -1..1);
The whole circle, however, is not a graph of a single function, and
it is more convenient to regard it as a parametric curve. That is, we
imagine sketching both x=cos(t) and y=sin(t) as coordinates as functions
of time:
> plot([cos(t), sin(t), t = -Pi..Pi]);
This worksheet will illustrate arc length from both points of view,
curves y = f(x) and parametric curves.
Arc length for a graph y = f(x)
To evaluate the length of the curve y = f(x) for x = a to x = b with
Maple, we may define the following procedure:
> fprime := D(f)(x);
x
fprime := - -----------
2 1/2
(1 - x )
> arclength := proc(f,a,b) int(sqrt(1 + fprime ^2), x = a..b) end;
Examples
Here is how to use this procedure for a specific function:
> f := 'f':
> f:= x -> sqrt(1 - x^2):
(Insert your own favorite function here. The previous command just
cleared any prior definition of f(x).)
Maple is able to use this procedure both with specific limits and
indefinite limits:
> arclength(f,0,1);
1/2 Pi
> evalf(");
1.570796327
> arclength(f,a,1);
Derivation
On a small scale, an arc of a smooth graph y = f(x) looks nearly straight,
like the hypotenuse of a right triangle. The horizontal side would have
length D x while the vertical side would have length D y. The arc
length D s along the hypotenuse is thus nearly sqrt( (D x)^2 + (D y)^2 ).
This is inconvenient, because if we wish to sum up infinitesimal arclengths
by integrating, we need an expression with a factor D x. To accomplish
this, we divide and multiply, obtaining sqrt( 1 + (D y /D x)^2 ) D x .
On the infinitesimal scale we can write the arc length element as:
ds = (sqrt(1 + (f'(x))^2) dx
The total arc length is approximated by the sum of the hypotenuses of
many small triangles fitting along the curve. When we pass to the limit
of infinitesimally fine partitions, we get the integral formula used above.
Observations
Like area, arc length can only be >= 0. If you perform the calculation
and you get a negative answer, you have made an error.
Also notice that sqrt(1 + fprime ^2) is always greater than or equal to 1.
This means that the integral for arc length for x = a to x = b will always
be at least b-a. Why is this? b-a is a straight-line distance between
the two values of x. The distance along the curve can never be less than
the straight-line distance b-a. If you perform the calculation and you
get something less than b-a, you have made an error.
It is actually rather rare that an arc length integral can be
done in closed form. When this is not possible, Maple returns an
expression which still involves an integral sign. If you follow this
expression with the command > evalf("), Maple will evaluate the
integral numerically (by a procedure similar to a Riemann sum, but more
sophisticated in order to be quicker).
Arc length and speed for a parametric curve
To evaluate the length of the curve given by x = x(t), y = y(t) for
t = a to x = b with Maple, we may define the following procedure:
> xprime := D(x)(t);
xprime := -sin(t)
> yprime := D(y)(t);
yprime := cos(t)
> arclengthp := proc(x,y,a,b) int(sqrt(xprime ^2 + yprime ^2), t = a..b) end;
arclengthp :=
proc(x, y, a, b) int(sqrt(xprime^2 + yprime^2), t = a .. b) end
Examples
Here is how to use this procedure for a specific function:
> x := 'x': y := 'y':
>
> x := t -> cos(t): y := t -> sin(t):
(Insert your own favorite functions here. The previous command just
cleared any prior definition of f(x).)
Maple is able to use this procedure both with specific limits and
indefinite limits:
> arclengthp(x,y,0,Pi);
Pi
> evalf(");
3.141592654
> arclengthp(x,y,a,b);
b - a
>
Derivation
On a small scale, an arc of a smooth graph y = f(x) looks nearly
straight, like the hypotenuse of a right triangle. The horizontal
side would have length D x while the vertical side would have length
D y. The arc length D s along the hypotenuse is thus nearly
sqrt( (D x)^2 + (D y)^2 ). This is inconvenient, because if we wish to
sum up infinitesimal arclengths by integrating, we need an expression
with a factor D t. To accomplish this, we divide and multiply,
obtaining sqrt( (D x /D t) + (D y /D t)^2 ) D t . On the
infinitesimal scale we can write the arc length element as:
ds = (x'(t)^2 + y'(t)^2 ) dt
The total arc length is approximated by the sum of the hypotenuses
of many small triangles fitting along the curve. When we pass to the
limit of infinitesimally fine partitions, we get the integral formula
used above.
Observations
Like area, arc length can only be >= 0. If you perform the calculation
and you get a negative answer, you have made an error.
We can interpret the expression (x'(t)^2 + y'(t)^2 ) as the speed. The
velocity vector would have components x' in the x-direction and y' in
the y-direction, and by Pythagoras's law, the square of its length is
the sum of the squares of its components
Exercises
y = f(x), integrals in closed form
1. Find the length of the curve y = x^(3/2) from (0,0) to (1,1)
2. Find the length of the curve y = x^3 /3 + 1/(4 * x) from x=1 to x=3.
y = f(x), integrals which must be evaluated numerically
1. Find the length of the curve y = e^x) from (0,1) to (1,e).
2. Find the length of the curve y = x^3 from x=1 to x=3.
Lengths of parametric curves
1. Find the length of the curve x = cos(t), y = t - sin(t) from t=0 to t=Pi/2.
2. The "astroid" is the curve defined by x = cos(t) ^3 , y = sin(t) ^3
Plot the curve for t=0 to t=Pi/2 and find its length.
Compare your calculation with the one for the nonparametric form of the curve,
x^(2/3) + y^(2/3) = 1.
3. Find the arc length between two points of the ellipse
x = cos(t), y = 4 * sin(t).
4. Find the distance traveled in the time t=0 to t=1 if
x = t^2, y = (2/3) (2 * t + 1)^(3/2)
Some problems involving speed
1. Find the speed at time t of a point on the ellipse
x = cos(t), y = 4 * sin(t).
2. A projectile tavels along the trajectory
x(t) = 5 * t, y(t) = 10 * t - 4.9 * t^2
What is its speed at time t? When and where is it traveling the slowest?
3. If an object at time t is at position
x= 6 * cos(t) + 5 * cos(3*t), y = 6 * sin(t) - 5 * sin(3*t),
it traces a curve called a hypotrochoid. Sketch the curve, and
determine the speed of the object when it crosses the x-axis.