Area under the Exponential Curve
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 /(k(-))=1.
(n/k,)
a
n
a
n
n
(1-1/k)
n=1,2,3,…
x
x
a
n
a
n
a
n+1
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.
x
k
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.
x=n/k
n
k
y=
-x
e
x