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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 iterated integral ...
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 ...
Laplace's method. In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form. where is a twice- differentiable function, is a large number, and the endpoints and could be infinite. This technique was originally presented in the book by Laplace (1774).
In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus. For example, many asymptotic expansions are derived from the ...
The integral can be reduced to a single integration by reversing the order of integration as shown in the right panel of the figure. To accomplish this interchange of variables, the strip of width dy is first integrated from the line x = y to the limit x = z , and then the result is integrated from y = a to y = z , resulting in:
This visualization also explains why integration by parts may help find the integral of an inverse function f−1 (x) when the integral of the function f (x) is known. Indeed, the functions x (y) and y (x) are inverses, and the integral ∫ x dy may be calculated as above from knowing the integral ∫ y dx.
The sequence () is decreasing and has positive terms. In fact, for all : >, because it is an integral of a non-negative continuous function which is not identically zero; + = + = () () >, again because the last integral is of a non-negative continuous function.
Both the Cauchy formula and the Riemann–Liouville integral are generalized to arbitrary dimensions by the Riesz potential. In fractional calculus, these formulae can be used to construct a differintegral, allowing one to differentiate or integrate a fractional number of times. Differentiating a fractional number of times can be accomplished ...
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