The emergence of the concept of the integral was due to the need to find the antiderivative function by its derivative, as well as to determine the amount of work, the area of complex figures, the distance traveled, with parameters outlined by curves described by nonlinear formulas.
From course
and physics knows 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 an integral of a constant? This is a linear function of the form y=kx+c.
But the force during the work can change, and in some kind of natural dependence. The same situation occurs with the calculation of the distance traveled if the speed is not constant.
So, it's clear what the integral is for. Its definition as the sum of the products of function values by an infinitesimal increment of the argument fully describes the main meaning of this concept as the area of a figure bounded from above by the line of the function, and at the edges by the boundaries of the definition.
Jean Gaston Darboux, French mathematician, in the second half of the XIXcentury very clearly explained what an integral is. He made it so clear that in general it would not be difficult even for a junior high school student to understand this issue.
Let's say there is a function of any complex form. The y-axis, on which the values of the argument are plotted, is divided into small intervals, ideally they are infinitely small, but since the concept of infinity is rather abstract, it is enough to imagine just small segments, the value of which is usually denoted by the Greek letter Δ (delta).
The function turned out to be "cut" into small bricks.
Each argument value corresponds to a point on the y-axis, on which the corresponding function values are plotted. But since the selected area has two borders, there will also be two values of the function, more and less.
The sum of the products of larger values by the increment Δ is called the large Darboux sum, and is denoted as S. Accordingly, the smaller values in a limited area, multiplied by Δ, all together form a small Darboux sum s. The section itself resembles a rectangular trapezoid, since the curvature of the line of the function with its infinitesimal increment can be neglected. The easiest way to find the area of such a geometric figure is to add the products of the larger and smaller value of the function by the Δ-increment and divide by two, that is, determine it as the arithmetic mean.
This is what the Darboux integral is:
s=Σf(x) Δ is a small amount;
S=Σf(x+Δ)Δ is a big sum.
So what is an integral? The area bounded by the function line and the definition boundaries will be:
∫f(x)dx={(S+s)/2} +c
That is, the arithmetic mean of large and small Darboux sums.c is a constant value that is set to zero 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 speed function, and limited by the time interval along the abscissa axis, will be the length of the path traveled.
L=∫f(x)dx on the interval from t1 to t2, Where
f(x) – speed function, that is, the formula by which it changes over time;
L – path length;
t1 – start time;
t2 – end time of the journey.
Exactly according to the same principle, the amount of work is determined, only the distance will be plotted along the abscissa, and the amount of force applied at each particular point will be plotted along the ordinate.
