The word "integral
Historical background of the emergence of the concept of integral
In the early 17th century in consideration of the leading scientists were a large number of physical (especially mechanical) task, in which it was necessary to investigate according to some variables from others. The most visible and pressing problems was the definition of the instantaneous velocity of nonuniform motion of a body in any time and reverse this, the problem of finding the values of the path traveled by the body over a period of time when such motion. Today we already know what the integral of speed
At first, from consideration of such dependences of physical quantities, for example, the path from speed, was formed mathematical concept of the function y = f(x). The study of the properties of various functions has led to the emergence of mathematical analysis. Scientists are actively looking for ways to study the properties of different functions.
How did the calculation of integrals and derivatives?
After creating Descartes fundamentals of analytic geometry and the appearance of the ability to represent functional dependencies graphically in the axes of a Cartesian coordinate system, the researchers had two major new challenges: draw a tangent line to the curve at any point and how to find the area of the figure bounded from above this curve and lines parallel to the coordinate axes. Unexpectedly it turned out that the first of them is equivalent to finding the instantaneous velocity and the second
The genius of Leibnitz and Newton in the mid 17th century the methods, allowing to solve both these problems. Was that for a tangent to the curve at the point you need to find the magnitude of the so-called derivative of the function describing this curve, in this point, and this value is equal to the rate of change of a function, i.e. with regard to dependence "the way of the speed
To find the area bounded by a curve, it was necessary to calculate a definite integral, which gave her the exact amount. Derivative and integral
The area under a curve
So, how to determine its exact value? Let us demonstrate the process of calculation through the integral in detail, from the very beginning.
Let f is continuous on a segment [ab] function. Consider the curve y = f(x) shown in the figure below. How to find the area of the region bounded curve ), x axis and lines x = a and x = b? That is the area of the hatched shape in the figure.
The simplest case, when f is a constant functionIn this case, the area under the curve is just a rectangle with height k and width (b
Area some other simple shapes such as triangle, trapezoid and semicircle are given by formulas from plane geometry.
The area under any continuous curve y = f(x) is given by a definite integral, which is written the same as a normal integral.
Before diving into a detailed answer to the question what the integral is, highlight some main ideas.
First, the area under the curve is divided into a number n of vertical strips small enough width
Drawing our rectangles width
In principle, it is possible to draw approximating rectangles so that the curve lay the rightmost point of the upper short sides of a width
But we can also take the height of each approximating rectangle equal to just a certain value of the function at an arbitrary point x*i inside the matched strip i (see Fig. below). We don't even have to take the same width of all the strips.
Draw up a Riemann sum:
The transition from the Riemann sum to a definite integral
In higher mathematics it is proved the theorem which says that if unlimited increase in the number n of rectangles approximating the greatest width tends to zero, then the Riemann sum of An tends to a limit A. the Number A i .
A clear explanation of the theorem is given by the picture below.
It can be seen that, the narrower the rectangles, the closer the area of a stepped shape to the area under the curve. When the number of rectangles ni n numerically equal to the desired square. This limit is the definite integral of the function f (x):
The symbol of the integral, which is a variation of the italic letter S, was introduced by Leibniz. To put on the top and bottom denote the integral of its limits suggested by J. B. Fourier. In this case clearly indicate the start and end x value.
The mechanical and geometrical interpretation of the definite integral
Try to give a detailed answer to the question about what is integral? Consider the integral on the interval [a,b] from the positive within it of the function f(x), and consider that the upper limit is more bottom a
If the ordinates of the function f(x) is negative in [a,b], then the absolute value of the integral equal to the area between the x-axis and the graph y=f(x), the integral is negative.
In the case of single or repeated intersection of the graph y=f(x) to the x-axis on the interval [a,b], as shown in the figure below, to calculate the integral we need to determine the difference in which the minuend is equal to the total area of plots above the abscissa axis, and the subtrahend So, for the function shown in the figure above, the definite integral from a to b is equal to (S1 S3)
Mechanical interpretation of the definite integral is closely related to the geometric. Back to the "Riemann sum
The total of the areas of rectangles over the interval from t1 =a to t2 =b will Express approximately the path s over time t2 1. and limit it, ie the integral (defined) from a to b of the function v = f(t) dt will give the exact value of path s.
The differential of a definite integral
If you go back to his designation, then it can be assumed that a = const and b is a particular value of some independent variable x. Then the definite integral with upper limit x of a specific number becomes a function of x. This integral is equal to the area of the shape under the curve, marked with dots aABb in the picture below. While fixed-line mobile aA and Bb, this area becomes a function f(x), and increment
Suppose that we gave the variable x = b for some small increment
From this we can conclude that the calculation of integrals is to razyskaniya functions according to the given expressions for their differentials. Integral calculus represents a system of methods of researches of such functions known to their differentials.
The fundamental relationship of integral calculus
It links the relationship between differentiation and integration and shows that there is an inverse operation of differentiation of functions – integrated. It also shows that if any function f(x) continuous, then by using this mathematical operation, you can find an ensemble (set, set) functions, the integral for it (or otherwise, find the indefinite integral of it).
Let the function F(x) is the designation of the result of integration of the function f(x). The correspondence between these two functions result from the integration of the second of them is designated as follows:
As can be seen in the symbol of integral lacking limits of integration. This means that it is transformed into a particular indefinite integral. The word "undefined
It should be emphasized that if the integral defined from a function is a number, then the indefinite is a function, or rather, a lot of them. The term "integration
The basic rule of integration
It represents the opposite of the corresponding rule for differentiation. How are indefinite integrals? Examples of this procedure we will look at the specific functions.
Let's look at the power function of the General form:
After we did this to each term in the expression integrable functions (if several), we add the constant at the end. Recall that taking the derivative of a constant is destroying her, so taking the integral of any function will give us the recovery of this constant. We denote it With a constant unknown – it could be any number! Therefore, we can have infinitely many expressions for the indefinite integral.
Let's consider the simple indefinite integrals, examples the taking of which is shown below.
May need to find the integral of:
f(x) = 4x 2 2x
Let's start with the first term. We look at the exponent of 2 and incrementing it by 1, then divide the first term on the resultant figure 3. We get: 4(x 3 ) / 3.
We then look at the next member and do the same thing. As it has exponent 1, the resulting figure will be 2. Thus, we divide that term by 2: 2(x 2 ) / 2 = x 2 .
The last term has a factor of x, but we just can't see it. We can imagine how the last term (-3x 0 ). This is equivalent to (-3)∙(1). If we use the integration rule, we will add 1 to the figure to raise it to first degree, and then divide the last term by 1. Get 3x.
This integration rule works for all values of n except n =
We have considered the simple example of finding the integral. In General, the solution of the integrals is not an easy task and good help is already gained in mathematics experience.
Table of integrals
In the section above, we saw that every differentiation formula is obtained the corresponding formula of integration. Therefore, all possible options have long been obtained and are summarized in the relevant tables. The following table of integrals contains formulas of integration of basic algebraic functions. These formulas need to know for memory, learning them gradually, as consolidation exercises.
Yet another table of integrals contains the basic trigonometric functions:
How to calculate definite integral
It turns out that to do this, being able to integrate, i.e. find the indefinite integral is very simple. And helps in this formula of the founders of the integro-differential calculus of Newton and Leibniz
According to her, the computation of the required integral is the first stage in finding undetermined, subsequent calculation of the integral was found F(x) by substituting x equal to the first upper limit, then lower, and finally, determining the difference between these values. While the constant can not record. because it is lost when performing the subtraction.
Take a look at some integrals with detailed solutions.
Find the area under one half wave of the sine wave.
Let us now consider the integrals with detailed solutions, using the additivity property in the first example, and using the intermediate integration variable in the second example. We calculate a definite integral of a fractional rational function:
Now show how you can simplify the capture of the integral by the introduction of an intermediate variable. May need to calculate the integral of (x-1) 2 .
About improper integrals
We talked about a definite integral for finite interval [a,b] of continuous functions on it, f(x). But a number of specific tasks leads to the necessity to expand the concept of the integral when the limits (one or both) is infinite, or discontinuous functions. For example, when calculating areas under the curves asymptotically approach the coordinate axes. For the extension of the concept of integral in this case, in addition to passing to the limit in the calculation of a Riemannian sum approximating rectangles is performed another. In this two-fold transition to the limit it turns the improper integral. In contrast, all the integrals mentioned above are called private.