Advanced Mathematics: Fifth Chapter definite integral (1) concept and property median theorem calculus Basic Formula _ Advanced Mathematics

Source: Internet
Author: User

The concept of §5.1 definite integral

First, from Archimedes ' exhaustion law

"Introduction" from the curve and straight line, the area of the surrounding figure.

For example: Insert a dividing point on the interval, get the point on the curve, over these points respectively to the axis, the axis of the vertical, to get the ladder shape. Their respective areas are:

Therefore, the area value can be

To facilitate the understanding of Archimedes ' thoughts, we first introduce the concept of curved edge trapezoid.

The so-called curved edge trapezoid refers to such a figure, it has three edges that are straight segments, where two are parallel, the third is perpendicular to the first two, and the fourth edge is a curved arc called a curved edge, which is at most one point to any line perpendicular to the bottom.

According to this definition, the area of the plot of the sample is a curved edge trapezoid area. Running program GS0501.M, can understand more profoundly the thought of Archimedes exhaustion law.

Area calculation of Curved edge trapezoid

Set the continuous function, to find the curved edge, straight line, and the axis of the curved edge of the trapezoidal area.

Insert a point arbitrarily in the interval, as shown in the diagram

The interval is divided into a small interval, and the length between the cells is

After each point is parallel to the axis of straight lines, these straight segments will be curved edge trapezoid divided into a narrow curved edge trapezoid, with the first narrow curved edge trapezoid area.

(due to the high on the curvature of the trapezoid is continuously changing, in a very short period of time its changes are very small, it can be approximately considered unchanged.) Therefore, in each small interval, can be used to approximate the height of a certain point to replace the small side of the plot of the trapezoid on the high change, with the corresponding little rectangular area to approximate the area of the trapezoid. )

In particular

The trapezoid of the first narrow curved edge is arbitrarily taken on its corresponding interval to approximate height and rectangular area.



Well, obviously.

The smaller the length of the cell, the better the approximate degree; the better the approximation, the less it is. Therefore, in order to get the exact value of the area, we only need to subdivide the interval infinitely, so that the length between each cell tends to zero.

If you remember, the length of each cell tends to be 0 of the price.

Thereby (1)

Three, the speed of linear movement of the distance

To set an object as a straight line motion, the known velocity is a continuous function on the interval, and the distance that the object passes through the time interval is obtained.

Arbitrarily insert a point in a time interval

will be divided into a time interval

The length of each time interval is

The distance of the movement of objects in each time interval is

At intervals, an approximation of the distance the object passes is

Namely: the speed of the object on the same as the constant, since the approximate substitution. Naturally, this approximation is reasonable when the time interval is short.

So the approximate values that can be given

In order to get the exact value, only the length of each small interval segment tends to be zero.

If you remember

The (2)

The above two examples, although their practical significance is different, but there are two points are consistent.

1. The area value of the curved edge trapezoid is determined by the high and changing range;

The distance of the speed line movement is determined by the velocity and the change interval.

2, the calculation and method, the same procedure, and all boils down to a structure exactly the same and type limit.

We can give the concept of definite integral by taking the concrete meaning of these problems into consideration and grasping their common essence in quantitative relation.

Iv. definition of Definite integral

Set a function to be bounded on, insert any points in

Divide the interval into small intervals

The length of each interval is

Take a little in each of the small intervals,

The product of the function value and the small interval length

Make and type


If no matter how the division of the interval,

And no matter how the dots on the cell,

As long as then, and always tending to a certain value,

We call this limit value as the definite integral of the function on the interval.

Recorded as


Which is called the integrand, called the integrable expression;

It's called an integral variable, called an integral interval.

Called the lower integral limit;

The integral and the formula called on.

If the definite integral is present, we can say that the upper integrable.

For the definition of definite integral, we give two important annotations:

1, the geometric meaning of definite integral

The area of a curved edge trapezoid formed by a curve, line, and axis.

On the top, the negative value of the trapezoid area of the curved edge is indicated.

Therefore, a definite integral is a numerical value.

2. The definite integral is independent of the integral variable

From the geometrical meaning of definite integral, we know:

The definite integral is related to the integrand and the integral interval.

If you do not change the integrand, and do not change the integral interval, but simply rewrite the variable to other letters, such as or, the value of the definite integral is still unchanged. that there

The existence theorem of definite integral

"Theorem One" is set on the interval continuous, then in the upper integrable.

"Theorem Two" is bounded on the interval, and only a finite discontinuity point, then the upper integrable.

Typical examples of definite integrals by definition

"Example 1" asks

Solution: It is continuous, so it exists.

In order to be convenient to calculate, divide the interval into equal parts, that is, to take points as

In this way, the length between the cells, and then take

The integral and the type are

Write an expression in a compact form:


This example tells us this information:

1, the definition of definite integral to calculate the definite integral is inconvenient, it is necessary to find a simple and effective method of calculation;

2, also reflects the accuracy of the geometric meaning of the definite integral.

The property and mean value theorem of §5.2 definite integral


1, when,

2, when,

The meaning of these two rules is more intuitive.

At that time, curved edge trapezoid shrink into a section of line, so its area should be zero;

At that time, the points corresponding to the interval became

The length between the corresponding cells.

At this point, the relative to the symbol should be the opposite.

Disclaimer: In the discussion below, the size of the upper and lower bounds of the integral is not limited, and the definite integrals listed in each property are assumed to exist.

The definite integral of the sum (difference) of the "property One" function equals the sum (difference) of the definite integral.



Obviously, a property is also established for any finite function.

The constant factor of the "nature two" integrand can refer to the integral number outside.

That is: (is the constant factor)


If the integral interval is divided into two parts, the definite integral on the whole interval equals the sum of the definite integrals on the two intervals.

namely: (*)

The geometrical meaning of this property is very obvious. As shown in the figure, the area of curved edge trapezoid has:

This property shows that the definite integral is additive to the integral interval. In fact, regardless of the relative position of three numbers, the equation (*) is always established.

For example: At that time, there

"Nature Four" if in the interval, then.

"Nature Five" if in the interval, then.

According to the definite integral geometric meaning, it is a curved edge trapezoid true area value, therefore it should be nonnegative.

"Inference One" if in the interval, then

In fact, by the nature of the five with the nature of a

"Inference Two"


By inference one has:


"Nature Six" is set and is the maximum and minimum value of the function on the interval.




This property can be used to estimate the range of the definite integral value, and it also has distinct geometrical significance.

"Nature Seven" (mean value theorem of definite integral)

If a function is contiguous on a closed interval, there is at least a little


Proof: According to the nature of six have

The numerical value is between the minimum and maximum value of the continuous function, and then the intermediate value theorem of the continuous function on the closed interval, which makes

Geometric interpretation of the mean value formula of integrals

Using computer to write program GS0502.M, to definite integral

We can verify the correctness of the mean value theorem of the definite integral by the numerical calculation experiment. When running the program, pay attention to establish the function file of the integrand f.m

Basic formula of §5.3 calculus

function and its derivative of the upper limit of integral

Set the function to be contiguous on the interval, and set it to the point above, and examine the integrals on the partial interval

This particular form of integral has two points that should be noted:

Firstly, the definite integral exists in the continuous. At this point, the variable "body and two posts", is not only the integral variable, but also the upper limit of the integral.

For the sake of clarity, the integration variable is converted to other symbols as the Tathagata, because the definite integral is independent of the selection of the integral variable. The definite integral above is rewritten as the following form

Secondly, if the upper limit is arbitrarily changed, it corresponds to each set, and the definite integral has a corresponding value. So, it defines a new function on the

A function called a variable with an integral upper limit (variable upper bound function).

Is there such a function?

To observe an example, the variable upper bound function on the normal curve is

It represents the area of a curved edge trapezoid. Run the program GS0503.M, can be made separately, in the image on the

This indicates that it is indeed a new function.

"Theorem One" if the function is contiguous on the interval, the upper bound function is changed

has a derivative on it, and its derivative is

Proof: When the upper limit is obtained increment, the function value at the place is

The increment of the resulting function

According to the integral mean value theorem:

In and between


Theorem One shows: Yes a primitive function. Therefore, we have the existence theorem of the following original function.

"Theorem Two" if the function is in the interval

Contact Us

The content source of this page is from Internet, which doesn't represent Alibaba Cloud's opinion; products and services mentioned on that page don't have any relationship with Alibaba Cloud. If the content of the page makes you feel confusing, please write us an email, we will handle the problem within 5 days after receiving your email.

If you find any instances of plagiarism from the community, please send an email to: and provide relevant evidence. A staff member will contact you within 5 working days.

A Free Trial That Lets You Build Big!

Start building with 50+ products and up to 12 months usage for Elastic Compute Service

  • Sales Support

    1 on 1 presale consultation

  • After-Sales Support

    24/7 Technical Support 6 Free Tickets per Quarter Faster Response

  • Alibaba Cloud offers highly flexible support services tailored to meet your exact needs.