Hyperbolic Cosine Area / ArcLength Ratio

This applet explores the relation between the area under a region of a hyperbolic cosine (cosh) function, and the length of that function's graph over the same region. It can be shown that this ratio is constant, and the constant is the parameter of the cosh function. Interestingly, a much simpler function with this property is a simple horizontal line, with a positive y-intercept. The cosh parameter "a" is adjusted with a slider, as are the start point and width of the integration and arc length interval. See the attached PDF for the math background for this applet.