Monday, September 28, 2009

Minimize the area between a curve and a horizontal line

Let f be continuous increasing function on the interval [a,b]. Find c so that the area of the region bounded by x=a, x=b, y=f(x), and y=c is a minimum.
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  1. DavidSeptember 29, 2009 at 1:26 PM

    The answer is c = f((a+b)/2).

    Since f is continuous and increasing, it has an inverse function g. By turning the graph sideways, we see that the area A(c) is given by the following equation.

    A(c) = int (f(a) to c) [g(x) - a] dx + int (c to f(b)) [b - g(x)] dx.

    Using the fundamental theorem of calculus,

    A'(c) = 2g(c) - a - b,

    so the minimum area occurs when A'(c) = 0, or when g(c) = (a+b)/2, which is equivalent to c = f((a+b)/2).

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