The emergence of the concept of integral was due to the need to find the antiderivative function along its derivative, as well as to determine the magnitude of the work, the area of complex figures, the distance traveled, with parameters outlined by curves described by non-linear formulas.
Of course
and
physicists know that work is equal to the product of force and distance. If all movement occurs at a constant speed or the distance is overcome with the application of the same force, then everything is clear, you just need to multiply them. What is the integral of a constant?
This is a linear function of the form y = kx + c.
But strength over the course of work can change, and in some regular dependence. The same situation arises with the calculation of the distance traveled if the speed is unstable.
So, it’s clear why the integral is needed. Defining it as the sum of the products of the values of the function by an infinitely small increment of the argument completely describes the main meaning of this concept as the area of the figure bounded above by the line of the function, and along the edges by the boundaries of the definition.
Jean Gaston Darboux, a French mathematician, in the second half of the 19th century very clearly explained what an integral is. He made it so understandable that, on the whole, it would not be difficult for even a junior high school student to understand this issue.
Suppose there is a function of any complex form. The ordinate axis, on which the argument values are laid out, is divided into small intervals, ideally they are infinitesimal, but since the concept of infinity is quite abstract, it is enough to imagine just small segments, the size of which is usually denoted by the Greek letter Δ (delta).
The function turned out to be “chopped” into small bricks.
Each value of the argument corresponds to a point on the ordinate axis, on which the corresponding values of the function are plotted. But since the boundaries of the selected section are two, then the values of the function will also be two, larger and smaller.
The sum of the products of the larger values by the increment Δ is called the large Darboux sum, and is denoted by S. Accordingly, the values smaller in the limited section multiplied by Δ together form the small Darboux sum s. The section itself resembles a rectangular trapezoid, since the curvature of the function line with its infinitely small increment can be neglected. The easiest way to find the area of such a geometric figure is to add the product of the larger and smaller values of the function by Δ-increment and divide by two, that is, define as the arithmetic mean.
Here is what Darb integral is:
s = Σf (x) Δ is a small sum;
S = Σf (x + Δ) Δ is a large sum.
So what is an integral? The area bounded by the line of the function and the boundaries of the determination will be equal to:
∫f (x) dx = {(S + s) / 2} + c
That is, the arithmetic mean of the large and small Darboux sums.c is a constant value, zeroed out during differentiation.
Based on the geometric expression of this concept, the physical meaning of the integral becomes clear. The area of the figure, outlined by the velocity function, and limited by the time interval along the abscissa axis, will be the length of the path traveled.
L = ∫f (x) dx in the interval from t1 to t2,
Where
f (x) is a function of speed, that is, the formula by which it changes in time;
L is the length of the path;
t1 is the start time of the path;
t2 is the end time of the path.
Exactly by the same principle, the magnitude of the work is determined, only the abscissa will be delayed, and by the ordinate the magnitude of the force applied at each particular point.