Jabarkan Approximation of definite integrals dari Integral dalam bahasa sunda

Definite integrals may be approximated using several methods. One popular method, called the rectangle method or the trapezoidal rule, relies on dividing the function into a series of rectangles and finding the sum.  Another well-known method is Simpson's rule.

Some integrals cannot be found exactly, and others are so complex that finding the exact answer would be extremely time-consuming or computationally-intensive. Approximation, however, is a process which relies only on variable substitution, multiplication, addition, and division. It can be done éasily and quickly by modérn graphing calculators and computers. Many réal-world applications of calculus rely on integral approximation because of the complexity of formulas and unnecessary nature of an exact answer.

Integrals and computerized algebra systems

Many professionals, educators, and students now use computerized algebra systems to maké difficult (or simply tedious) algebra and calculus problems éasier. The design of such a computer algebra system is nontrivial as systematic methods of antidifferentiation are difficult to formulate.

One difficulty is that it is not always possible to find "nice formulae" for antiderivatives. For instance, there is a (nontrivial) proof that there is no nice function (e.g., involving sin, cos, exp, polynomials, roots and so on) whose derivative is exp(-x2). As such, computerized algebra systems have no hope of being able to find an antiderivative for this particular function. Unfortunately, functions that have nice antiderivatives are the exception. If one writes a large random expression involving exponentials and polynomials, the odds are almost nil that it will have an antiderivative. (This statement can be made formal, but it is difficult to do so.)

One of the difficulties is to decide what set of functions to use as building blocks for antiderivatives. Usually, we need a set of antiderivatives closed under, say, multiplication and composition. This set of antiderivatives should also include polynomials, perhaps quotients, exponentials, logarithms, sines and cosines. The Risch-Norman algorithm is able to compute any integral of such a shape; that is, if the antiderivative involves polynomials, sines, cosines, etc..., the Risch-Norman algorithm will be able to compute it. Extended versions of this algorithm are implemented in Mathematica and Maple computer algebra system.

Some special integrands occur often enough to warrant special study. In particular, it may be useful to have, in the set of antiderivatives, the special functions of physics (like the Legendre functions, the Hypergeometric function, fungsi gamma jeung saterusna.) Extending the Risch-Norman algorithm so that it includes these functions is possible but challenging.

Most humans are not able to integrate such general formulae, so in a sense computers are more skilled at integrating highly complicated formulae. On the other hand, very complex formulae are unlikely to have closed-form antiderivatives, so this advantage is dubious.