What the geometrical meaning of a certain integral consists in

What the geometrical meaning of a certain integral consists in

Many mathematical concepts and especially method of the mathematical analysis seem absolutely abstract and unsuitable for real life. But it anything else, as delusion of the layman. Not for nothing to the mathematician nicknamed the queen of all sciences.

Itself cannot imagine the modern mathematical analysis without application of a concept of integral and methods of integral calculus. In particular, a certain integral strongly was fixed not only in mathematics, but also in physics, mechanics and many other scientific disciplines. The concept of integration is opposite to differentiation and means association of parts, for example, of any figure in whole.

History of a certain integral

Methods of integration originate in antiquity. They were known in Ancient Egypt. There are facts demonstrating that the formula of volume of the truncated pyramid was known B.C. to Egyptians in 1800. She also allowed them to create such masterpieces of architecture as the Egyptian pyramids.

Initially integrals paid off by method of exhaustion Evdoks. Already at the time of Archimedes by means of integral calculus by the advanced Evdoksa method counted the areas of a parabola and a circle. The modern concept of a certain integral and a method were entered by Jean Baptiste Joseph Fourier approximately in 1820.

Concept of a certain integral and its geometrical sense

Without use of mathematical characters and formulas a certain integral can be designated as the sum of the parts making the geometrical figure formed by a curve of a concrete function graph. When it comes to a certain integral of the f (x) function, it is necessary to present this function in the system of coordinates at once. Such function in the form of a curve of abscissa axis stretching lengthways, that is an axis of X, at a certain distance from ordinate axis, that is an axis of Y will look. When you count integral ∫, you limit at first the received curve on axis x. That is you define from what and on what moment of an axis X you will consider this function graph of f (x). Visually you draw the vertical lines connecting a curve of the schedule and an axis X in the chosen points. Thus, under a curve the geometrical figure reminding a trapeze is formed. It is limited by the lines drawn by you at the left and on the right, from below it is framed with an axis of X, and from above – the most curve graphics. The received figure carries the name of a curvilinear trapezoid. To count the area S such difficult figures, use a certain integral. Certain integral of the f (x) function on the chosen piece on an axis of X allows to calculate without effort the area of a curvilinear trapezoid under a schedule curve. Its geometrical sense also consists in it.

Author: «MirrorInfo» Dream Team