Area Under the Tractrix

slices

align

tractrix

The tractrix is a curve such that any tangent segment from the tangent point on the curve to its asymptote (the

x

axis) has constant length

a

. Using this definition, slices of a disk are placed in contact with each other and the

x

axis. As you increase the number of slices, their envelope tends to the tractrix. When you use the "align" slider, these slices form a half-disk. Therefore, the area under the tractrix is equal to the area of a half-disk with radius

a

, which is equal to

π

2

a

/2

.

Details

For more information visit "Visual Calculus by Mamikon".

T. M. Apostol and M. Mamikon, "Subtangents—An Aid to Visual Calculus," American Mathematical Monthly, 109(6), 2002 pp. 525–533.

T. M. Apostol and M. Mamikon, "Tangents and Subtangents Used to Calculate Areas," American Mathematical Monthly, 109(10), 2002 pp. 900–908.

External Links

Tractrix (Wolfram MathWorld)

Permanent Citation

Okay Arik

"Area Under the Tractrix"
http://demonstrations.wolfram.com/AreaUnderTheTractrix/
Wolfram Demonstrations Project
Published: January 28, 2008