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Integration is the basic operation in integral calculus.While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful.
In mathematics, the definite integral ∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx} is the area of the region in the xy -plane bounded by the graph of f , the x -axis, and the lines x = a and x = b , such that area above the x -axis adds to the total, and that below the x -axis subtracts from the total.
Third kind: An integral equation is called an integral equation of the third kind if it is a linear Integral equation of the following form: [3] () + (,) = where g(t) vanishes at least once in the interval [a,b] [4] [5] or where g(t) vanishes at a finite number of points in (a,b).
A line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve. [42] Various different line integrals are in use. In the case of a closed curve it is also called a contour integral. The function to be integrated may be a scalar field or a vector field.
Taking the difference of each side between two values = and = and applying the fundamental theorem of calculus gives the definite integral version: ′ = () () ′ (). The original integral ∫ u v ′ d x {\displaystyle \int uv'\,dx} contains the derivative v' ; to apply the theorem, one must find v , the antiderivative of v' , then evaluate ...
In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing whether an infinite series or an improper integral converges or diverges by comparing the series or integral to one whose convergence properties are known.
The problem of the differentiation of integrals is much harder in an infinite-dimensional setting. Consider a separable Hilbert space (H, , ) equipped with a Gaussian measure γ. As stated in the article on the Vitali covering theorem, the Vitali covering theorem fails for Gaussian measures on infinite-dimensional Hilbert spaces. Two results of ...
is the fractional derivative (if q > 0) or fractional integral (if q < 0). If q = 0, then the q -th differintegral of a function is the function itself. In the context of fractional integration and differentiation, there are several definitions of the differintegral.