![]() ![]() ![]() However, for the purpose of this article we will stick to two-point quadrature.Īs stated in the first equation, the integral is estimated by evaluating f(x) at two points: x x1 and x x2 and multiplying it by two constants c1 and c2. Note that the number of points can be much larger the more points the more accurate the estimation. The beauty of Gaussian integration is that is does not fix the evaluation points at a fixed fraction of the interval, but rather it selects them at an optimum point (or points) in the interval.įigure 1 shows the two-point Gaussian quadrature for example. Recently, I learned about a class of methods knows as Gaussin Quadrature that fit the bill perfectly: simple and quite accurate for the effort involved Gaussian Quadrature Methods such as the Simpson rule estimate an integral by evaluating a function f(x) at a point equidistant from a and b in each interval a,b.įor example, with the Simpson rule, the function needs to be evaluated at x (ab)2 which is exactly the middle of the interval. Its an improvement over the trapezoidal method but not that accurate either. Most people learn about the Simpson rule for example in an introductory numerical anlysis class. It may be OK for a first-order estimate, but the accuracy can be pretty bad as demonstrated in the example further below.Įxponential Function Graph This is especially true when trying to integrate exponential functions which as luck would have it, are extremely common in Engineering problems (and nature in general).įortunately, there are many alternative methods of integration which are much more precise and are not very computationally intensive either. Simple as the method is, its also the leat accurate of them all. So the integral in the interval a,b can be simply estimated by the trapezoid area given by: S (b-a). The simplest of the methods is the so called trapezoidal rule integration.Īs the name indicates, we evaluate the function f(x) at a number of points and calaculate the total area (integral) as the sum of the areas of small trapezoids between two points. In many Engineering projects, one often needs to approximate integrals of a continuous function (think calculating power or energy). ![]() ![]() However, integrating an arbitrary continuous function in Excel is not one of the built-in function. The built-in library of mathematical functions cover many of the needs in day-to-day design task. ![]()
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