Area Under the Tractrix
The tractrix is a curve such that any tangent segment from the tangent point on the curve to its asymptote (the axis) has constant length . Using this definition, slices of a disk are placed in contact with each other and the 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 , which is equal to .
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.