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

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.

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


Practise This Question

Ram drew a line segment AB of length 3cm and another line segment CB of length 4cm which is perpendicular to line segment AB. What is the length of the third side of the triangle?