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

.