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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.
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.
The former expression is written as a definite integral and the latter is written as an indefinite integral. Applying the appropriate limits to the latter expression should yield the former, but the latter is not necessarily equivalent to the former. Mathematician Brook Taylor discovered integration by parts, first publishing the idea in 1715.
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature (often abbreviated to quadrature ) is more or less a synonym for "numerical integration", especially as applied to one-dimensional integrals.
calculate definite integral (also called solid of revolution) - finding volume by rotation (around the x-axis): int(pi*function^2,x,lowerlimit,upperlimit) calculate definite integral (also called solid of revolution) - finding volume by rotation (around the y-axis) for a decreasing function: int(2*pi*x*function,x,lowerlimit,upperlimit)
In numerical analysis, Romberg's method [1] is used to estimate the definite integral by applying Richardson extrapolation [2] repeatedly on the trapezium rule or the rectangle rule (midpoint rule). The estimates generate a triangular array .
For instance, when evaluating definite integrals using the fundamental theorem of calculus, the constant of integration can be ignored as it will always cancel with itself. However, different methods of computation of indefinite integrals can result in multiple resulting antiderivatives, each implicitly containing different constants of ...
In calculus, symbolic integration is the problem of finding a formula for the antiderivative, or indefinite integral, of a given function f(x), i.e. to find a formula for a differentiable function F(x) such that