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Area under the Exponential Curve

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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 .

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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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Contributed by: Okay Arik (March 2011)
Open content licensed under CC BY-NC-SA

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For more information visit "Visual Calculus by Mamikon".

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