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In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in (the real-number plane) are called double integrals, and integrals of a function of three variables over a region ...
Fubini's theorem. In mathematical analysis, Fubini's theorem characterizes the conditions under which it is possible to compute a double integral by using an iterated integral. It was introduced by Guido Fubini in 1907. The theorem states that if a function is Lebesgue integrable on a rectangle , then one can evaluate the double integral as an ...
An illustration of Monte Carlo integration. In this example, the domain D is the inner circle and the domain E is the square. Because the square's area (4) can be easily calculated, the area of the circle (π*1.0 2) can be estimated by the ratio (0.8) of the points inside the circle (40) to the total number of points (50), yielding an approximation for the circle's area of 4*0.8 = 3.2 ≈ π.
Integration by parts is a heuristic rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate this single function into a product of two functions u(x)v(x) such that the residual integral from the integration by parts formula is easier to evaluate than the ...
Double integrator. Feedback system with a PD controller and a double integrator plant. In systems and control theory, the double integrator is a canonical example of a second-order control system. [1] It models the dynamics of a simple mass in one-dimensional space under the effect of a time-varying force input .
The integration formula for double integrals may be generalized to any multiple integral. In all cases, there is a parameter value n ∗ {\textstyle n^{\ast }} or array of parameter values N ∗ {\textstyle N^{\ast }} that solves one or more linear equations derived from the exponent terms of the integrand's series expansion.
Calculus. In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, [1] is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can loosely be thought of as using the chain rule "backwards."
This simplifies the theory and algorithms considerably. The problem of evaluating integrals is thus best studied in its own right. Conversely, the term "quadrature" may also be used for the solution of differential equations: "solving by quadrature" or "reduction to quadrature" means expressing its solution in terms of integrals.