Area under the Exponential Curve

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.