 # Continuous Integration

By bringing together minuscule data integrals assign a number to a function in such a manner that describes volume, area and displacement. The two major operations of calculus are integration and its opposite that is differentiation.

Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral Continuous Integration

The connection between differential and integral calculus is through the fundamental theorem of calculus. In particular, if f(x) is continuous in the interval a <= x <=b and G(x) is a function such that (dG)/(dx) = f(x) for all values of x in (a, b), then

Let f be continuous on an interval I. Choose a point a in I. Then the function F(x) defined by

$\large F(x)=\int_{a}^{x}f(t)\;dt$

Let c be in I, and let x be infinitely close to c and between the endpoints of I. By the Addition Property,

$\large \int_{a}^{c}f(t)\;dt=\int_{a}^{x}f(t)\;dt+\int_{x}^{c}f(t)\;dt,$

$\large \int_{a}^{c}f(t)\;dt-\int_{a}^{x}f(t)\;dt+\int_{x}^{c}f(t)\;dt,$

$\large f(c)-F(x)=\int_{x}^{c}f(t)\;dt.$

Example to find out is continuous integration:

In what is continuous integration, let f(y) = in y, u(α) = a and v(α)=α

In this case, the function doesn’t depend on α Consequently, it has to be substitute with u, v, and f.

$\frac{(d)}{(d \alpha)}\int_a^{\alpha}\;In,y,dy= \frac{(f\alpha)(d, a)}{(d\;\alpha)}$