22 arc length

Archytas lived in southern Italy during the time of

Greek control. The region, then called Magna Graecia,

included the town of Tarentum, which was home to

members of the Pythagorean sect. Like the Pythagore-

ans, Archytas believed that mathematics provided the

path to understanding all things. However, much to

the disgust of the Pythagoreans, Archytas applied his

mathematical skills to solve practical problems. He is

sometimes referred to as the Founder of Mechanics

and is said to have invented several innovative mech-

anical devices, including a mechanical bird and an

innovative child’s rattle.

Only fragments of Archytas’s original work survive

today, and we learn of his mathematics today chiefly

through the writings of later scholars. Many results

established by Archytas appear in E

UCLID

’s famous text

T

HE

E

LEMENTS

, for instance.

arc length To measure the length of a curved path,

one could simply lay a length of string along the path,

pull it straight, and measure its length. This determines

the arc length of the path. In mathematics, if the curve in

question is continuous and is given by a formula y= f(x),

for a< x< bsay, then

INTEGRAL CALCULUS

can be used

to find the arc length of the curve. To establish this, first

choose a number of points (x1, y1),…,(xn, yn) along the

curve and sum the lengths of the straight-line segments

between them. Using the

DISTANCE FORMULA

, this gives

an approximate value for the length sof the curve:

Rewriting yields:

The

MEAN

-

VALUE THEOREM

shows that for each ithere is

a value cibetween xi–1 and xiso that ,

and so the length of the curve is well approximated by

the formula:

Of course, taking more and more points along the curve

gives better and better approximations. In the limit,

then, the true length of the curve is given by the formula:

This is precisely the formula for the integral of the

function over the domain in question.

Thus we have:

The arc length of a continuous curve y = f(x) over

the interval [a,b] is given by

Alternatively, if the continuous curve is given by a set of

PARAMETRIC EQUATIONS

x= x(t) and y= y(t),for a< t<

bsay, then choosing a collection of points along the

curve, given by t1,…,tnsay, making an approximation to

the curve’s length, and taking a limit yields the formula:

In a similar way one can show that if the continuous

curve is given in

POLAR COORDINATES

by formulae and

x= r(θ)cos(θ) and y= r(θ)sin(θ), for a< θ< b, then the

arc length of the curve is given by:

The presence of square-root signs in the integrands

often makes these integrals very difficult, if not impos-

sible, to solve. In practice, one must use numerical tech-

niques to approximate integrals such as these.

See also

NUMERICAL INTEGRATION

.

area Loosely speaking, the area of a geometric figure

is the amount of space it occupies. Such a definition