emergence of the concept of integral was caused by the necessity of finding a primitive function from its derivative, as well as determining the value of the work, the area of complex shapes, the distance traveled by the way, with the parameters outlined in the curves describing nonlinear equations.
From physics course known that the work is the product of force over a distance.If all motion at a constant velocity or the distance is overcome by application of the same force, the understanding, they need to simply multiply.What is the integral of the constants?This is a linear function of the form y = kx + c.
But the power over the operation may vary, and in some legitimate addiction.A similar situation arises with the calculation of distance, if the speed is not constant.
So, it is understandable why there is integral.Defining it as a sum of products of values of an infinitely small increment argument completely describes the principal meaning of the term as the area of the figure bounded by the top line functions, and the edges - the detection limit.
Jean Gaston Darboux, French mathematician, in the second half of the XIX century very clearly explained that this integral.He made it so clear that, in general understand this question is not difficult, even student junior high school.
Suppose there is a function of any complex shape.The y-axis on which the deposited value of the argument, is divided into small intervals, ideally, they are infinitely small, but because the concept of infinity is quite abstract, it is enough to imagine just small pieces, the size of which is usually denoted by the Greek letter Δ (delta).
function was "chopped" into smaller blocks.
each value argument corresponds to a point on the y-axis on which are deposited the corresponding values of the function.But as the boundaries of the selected area from the two, then the values of the function will also be two, more or less.
sum of the products of large values in the increment of Δ is called a large sum of Darboux, and is denoted as S. Accordingly, the smaller the values of a limited area, multiplied by Δ, together form a small amount of Darboux s.The site itself resembles a rectangular trapezoid, as the curvature of the line features an infinitesimal increment it can be neglected.The easiest way to find the area of a geometric figure - is to lay down a work of larger and smaller values of the function on Δ-increment and divide by two, that is defined as the arithmetic mean.
That is what the integral Darboux:
s = Σf (x) Δ - a small amount;
S = Σf (x + Δ) Δ - a large sum.
So, what is the integral?The area bounded by a line function and the detection limit will be equal to:
∫f (x) dx = {(S + s) / 2} + c
That is the arithmetic mean of major and minor amounts Darbu.s - constant,reset during differentiation.
Based on the geometric expression of this concept, it is clear, and the physical meaning of the integral.Square shapes, outlined a function of speed, and the limited time interval on the horizontal axis, will be the length of the distance traveled.
L = ∫f (x) dx in the interval t1 to t2,
Where
f (x) - a function of speed, that is the formula by which it changes over time;
L - length of the path;
t1 - time of beginning of the path;
t2 - time of the end path.
Exactly the same principle is determined by the amount of work only to be deposited in the abscissa the distance and the ordinate - the amount of force exerted in each point.
