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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.
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.
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form () (,), where < (), < and the integrands are functions dependent on , the derivative of this integral is expressible as (() (,)) = (, ()) (, ()) + () (,) where the partial derivative indicates that inside the integral, only the ...
Thus, the integral of the velocity function (the derivative of position) computes how far the car has traveled (the net change in position). The first fundamental theorem says that the value of any function is the rate of change (the derivative) of its integral from a fixed starting point up to any chosen end point.
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.
The slope field of () = +, showing three of the infinitely many solutions that can be produced by varying the arbitrary constant c.. In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral [Note 1] of a continuous function f is a differentiable function F whose derivative is equal to the original function f.
In calculus and mathematical analysis the limits of integration (or bounds of integration) of the integral () of a Riemann integrable function f {\displaystyle f} defined on a closed and bounded interval are the real numbers a {\displaystyle a} and b {\displaystyle b} , in which a {\displaystyle a} is called the lower limit and b {\displaystyle ...
A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that e −x 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.