Means Generated by an Integral

1. the arithmetic mean: A(a, b) = (a + b)/2 2. the geometric mean: G(a, b) = -b 3. the harmonic mean: H(a, b) = 2ab/(a + b) 4. the logarithmic mean: L(a, b) = (b a)/(ln b In a) 5. the Heronian mean: N(a, b) = (a + Na-b + b)/3 6. the centroidal mean: T(a, b) = 2(a2 + ab + b2)/3(a + b) Recently, Professor Howard Eves [1] showed how many of these means occur in geometrical figures. The integral in our title is