By Henry Ahner
Integration (here visualized as a pounding process) is mathematically realized by simple transformations, successively smoothing the bounding curve into a straight line and the region-to-be-integrated into an area-equivalent rectangle. The relationship to Riemann sums, and to the trapezoid and midpoint methods of numerical integration, is illustrated.
Trick or Technique?
By Michael Sheard
More often than one might at first imagine, a simple trick involving integration by parts can be used to compute indefinite integrals in unexpected and amusing ways. A systematic look at the trick illuminates the question of whether the trick is useful enough to be called an actual technique of integration.
By Vaughan Pratt
The inter-derivability of the Pythagorean Theorem and Heronís area formula is explained by applying Al-Karkhiís factorization to Heronís formula.
Diametric Quadrilaterals with Two Equal Sides
By Raymond A. Beauregard
If you take a circle with a horizontal diameter and mark off any two points on the circumference above the diameter, then these two points together with the end points of the diameter form the vertices of a cyclic quadrilateral with the diameter as one of the sides. We refer to the quadrilaterals in question as diametric. In this note we consider diametric quadrilaterals having two equal sides. They can be reduced to the two forms (a, b, a, d) (an isosceles trapezoid) and (b, a, a, d) (a skewed kite). It is shown that these quadrilaterals are diametric if and only if (r, a, d) is a right triangle satisfying d< √2a where r = √(d2 - a2), in which case the area of either quadrilateral is ((b+d)/4) √(d2- b2), where b = d - 2a2/d. The integer case (Brahmaguta quadrilaterals) is then considered. It is shown that each primitive Pythagorean triple (t, u, v) with v > √2u determines a Brahmagupta diametric quadrilateral with two equal sides uv, two other sides v2 – 2u2, v2, and area ut3. Moreover every primitive diametric quadrilateral with integer sides, two of which are equal, and with integer diagonals, arises in this way.
Solomonís Sea and π
By Andrew J. Simoson
This paper is a whimsical survey of the various explanations which might account for the biblical passage in I Kings 7:23 that describes a round objectóa bronze basin called Solomon's Seaóas having diameter ten cubits and circumference thirty cubits. Can the biblical pi be any number other than 3? We offer seven different perspectives on this historical puzzle.
Evolutionary Stability in the Travelerís Dilemma
By Andrew T. Barker
The traveler's dilemma is a generalization of the prisoner's dilemma which shows clearly a paradox of game theory. In the traveler's dilemma, the strategy chosen by analysis and theory seems obviously wrong intuitively. Here we develop a measure of evolutionary stability and show that the evolutionarily stable equilibrium is in some sense not very stable. Understanding evolutionary stability on a continuum instead of as a yes-or-no definition helps to explain and partially resolve the paradox in the traveler's dilemma.
Series involving iterated logarithms
By J. Marshall Ash
The boundary between convergent and divergent series is systematically explored through sums of iterated logarithms.
Sums of Integer Powers via the Stolz-Cesŗro Theorem
By Sidney H. Kung
The Stoltz-Cesŗro Theorem, a discrete version of líHŰpitalís rule, is applied to the summation of integer powers.
Short Division of Polynomials
By Li Zhou
The familiar process of synthetic division is extended to much more complicated divisors and turns out, surprisingly, not to be as difficult as one might imagine.
MOODY’S MEGA MATH CHALLENGE: A MODELING COMPETITION
PROBLEMS AND SOLUTIONS
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