Consider a curve consisting of segments joining the points
, where
and
. The region under this curve is broken into triangular pieces by extending the segments to the
axis. Each extended segment projects onto a segment of length 1 on the
axis because
.
You can align these triangles one on top of the other above the interval [0,1] on the
axis using the "align" slider. You can control the constant
using the "triangles per unit length" slider.
Let
. As
and
tend to infinity, the curve approaches the exponential curve
. The "total length" slider controls the length of the
interval. As the total length tends to infinity, the aligned triangles fill the unit square of area 1.
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