Calculating the volume of the figure limited lines online. Calculating the volume of rotation bodies using a specific integral

Subject: "Calculating the volume of rotation bodies using a specific integral"

Type of lesson:combined.

The purpose of the lesson: Learn to calculate the volume of rotation bodies using the integrals.

Tasks:

consolidate the ability to highlight curvilinear trapezes from a row geometric figures and to work out the skill of calculations of the area of \u200b\u200bcurvilinear trapeats;

get acquainted with the concept of a bulk figure;

learn to calculate the volume of the bodies of rotation;

promote development logical thinking, competent mathematical speech, accuracy in the construction of drawings;

brief interest in the subject, to the operating of mathematical concepts and images, to raise the will, independence, perseverance when the final result is reached.

During the classes

I. Organizational moment.

Greeting group. Message to students' targets.

Today's lesson would like to start with parables. "Sage lived, who knew everything. One person wanted to prove that the sage knows not all. Close in the palms of the butterfly, he asked: "Tell me, a sage, what a butterfly in my hands: Dead or Live?" And he thinks: "Says live - I will be dead, says dead - I will release." Sage, thinking, replied: "All in your hands."

Therefore, let's work fruitfully today, we will purchase a new luggage of knowledge, and the received skills and skills will be applied in future life and in practical activity. "Everything is in your hands."

II. Repetition of the previously studied material.

Let's remember the main points of the previously studied material. To do this, perform the task "exclude the unnecessary word".

(Students say an excess word.)

Right "Differential". Try the remaining words to call one common word. (Integral calculus.)

Let's remember the main stages and concepts associated with integral calculus ..

The task. Restore skip. (The student comes out and enters the marker necessary words.)

Work in notebooks.

Newton Newton's English physicist (1643-1727) and German philosopher Gottfried Leibnitsa (1646-1716) brought the formula Newton-Labitsa (1643-1727). And this is not surprising, because mathematics is a language that nature speaks.

Consider how when solving practical tasks This formula is used.

Example 1: Calculate the area of \u200b\u200bthe figure, limited lines

Decision: We construct on the coordinate plane of graphics of functions . We highlight the area of \u200b\u200bthe figure, which you need to find.

III. Studying a new material.

Pay attention to the screen. What is shown in the first drawing? (Flat figure is presented in the figure.)

What is shown in the second drawing? Is this figure flat? (The figure shows a bulk figure.)

In space, on earth and in everyday life We meet not only with flat figures, but also in bulk, but how to calculate the volume of such tel? For example: the volume of the planet, comet, meteorite, etc.

The volume of houses are conceived by the volume, and overflow water from one vessel to another. Rules and techniques for calculating volumes were to arise, another thing is how accurates they were and justified.

1612 was for residents of the Austrian city of Linz, where he lived then by astronomer Johann Kepler was very harvest, especially on grapes. People prepared wine barrels and wanted to know how to practically determine their volumes.

Thus, the considered works of Kepler marked the beginning of a whole flow of studies that were crowned in the last quarter of the XVII century. Registration in the works of I. Newton and G.V. Differential and integral leibher. Mathematics of variables of magnitude occupied from this time the leading place in the system of mathematical knowledge.

Today we are with you and we will deal with such practical activities, hence,

The theme of our lesson: "Calculation of rotation bodies using a specific integral".

You will learn the definition of the body of rotation by following the following task.

"Labyrinth".

The task. Find the output from the confused position and write down the definition.

IV Calculation of volumes.

Using a specific integral, it is possible to calculate the volume of a body, in particular, the bodies of rotation.

The body of rotation is called the body obtained by the rotation of the curvilinear trapezium around its base (Fig. 1, 2)

The volume of the body of rotation is calculated by one of the formulas:

1. around the axis oh.

2. if the rotation of the curvilinear trapezium around the OU axis.

Students write down the basic formulas in the notebook ..

The teacher explains the solution of examples on the board.

1. Find the volume of the body obtained by rotation around the axis of the ordinate of a curvilinear trapezoid, limited lines:x2 + y2 \u003d 64, y \u003d -5, y \u003d 5, x \u003d 0.

Decision.

Answer: 1163 cm3.

2. Find the volume of the body obtained by the rotation of the parabolic trapezium, around the abscissa axis y \u003d, x \u003d 4, y \u003d 0.

Decision.

V.. Mathematical simulator.

2. The combination of all the primary from this function is called

BUT) uncertain integral,

B) function,

C) differentiation.

7. Find the volume of the body obtained by rotation around the abscissa axis of the curvilinear trapezium limited to the lines:

D / s. Fastening a new material

Calculate body volume educated by rotation Petal, around the abscissa axisy \u003d x2, y2 \u003d x.

We construct graphs of the function. y \u003d x2, y2 \u003d x. Graph y2 \u003d x We convert to the type y \u003d.

We have V \u003d V1 - V2 calculate the volume of each function:

Output:

A certain integral is some foundation for learning mathematics, which makes an indispensable contribution to solving the tasks of practical content.

The topic "Integral" brightly demonstrates the connection of mathematics with physics, biology, economy and technology.

Development modern science unthinkable without using the integral. In this regard, it is necessary to start studying in the framework of secondary special education!

VI. Estimation.(Commenting.)

Great Omar Khayam - mathematician, poet, philosopher. He calls to be masters of his fate. We listen to the excerpt from his work:

You will say, this life is one moment.
Her appreciation, in her draws inspiration.
How to spend it and pass.
Do not forget: she is your creation.

Use of integrals to find the volume of bodies of rotation

The practical utility of mathematics is due to the fact that without

specific mathematical knowledge is hampered by the understanding of the principles of the device and use modern technology. Every person in his life has to do enough complex calculations, to use common techniques, find in reference books to apply the necessary formulas, to make simple algorithms for solving problems. IN modern society More specialties requiring high level Education is associated with the direct use of mathematics. Thus, for a schoolboy, mathematics becomes a professional meaningful subject. The leading role belongs to mathematics in the formation of algorithmic thinking, brings up the ability to act according to a given algorithm and design new algorithms.

Studying the topic of application of the integral to calculate the volume of bodies of rotation, I offer students at optional classes to consider the topic: "The volumes of rotation bodies with the use of integrals." Below, we give methodical recommendations for consideration of this topic:

1. Flat shape location.

From the course of algebra, we know that the concept of a certain integral has led to a practical task ... "Width \u003d" 88 "Height \u003d" 51 "\u003e. Jpg" width \u003d "526" height \u003d "262 src \u003d"\u003e

https://pandia.ru/text/77/502/images/image006_95.gif "width \u003d" 127 "height \u003d" 25 src \u003d "\u003e.

To find the volume of the body of rotation formed by the rotation of the curvilinear trapezium around the OX axis bounded by the interrupt line y \u003d f (x), the OX axis, direct x \u003d a and x \u003d b calculated by the formula

https://pandia.ru/text/77/502/images/image008_26.jpg "width \u003d" 352 "height \u003d" 283 src \u003d "\u003e y

3. Cylinder volume.

https://pandia.ru/text/77/502/images/image011_58.gif "width \u003d" 85 "height \u003d" 51 "\u003e .. gif" width \u003d "13" height \u003d "25"\u003e .. jpg " width \u003d "401" height \u003d "355"\u003e cone is obtained by rotation rectangular triangle ABC (C \u003d 90) around the OX axis on which the speaker is lying.

Cut AV lies on a straight line y \u003d kx + C, where https://pandia.ru/text/77/502/images/image019_33.gif "width \u003d" 59 "height \u003d" 41 src \u003d "\u003e.

Let a \u003d 0, b \u003d h (the height of the cone), then vhttps: //pandia.ru/text/77/502/images/image021_27.gif "width \u003d" 13 "height \u003d" 23 src \u003d "\u003e.

5. Punction of a truncated cone.

The truncated cone can be obtained by rotating the rectangular trapezium of the AVD (CDOX) around the OX axis.

Cut AB lies on a straight line y \u003d kx + C, where , C \u003d R.

Since the straight line passes through the point A (0; R).

Thus, the line has the appearance of https://pandia.ru/text/77/502/images/image027_17.gif "width \u003d" 303 "height \u003d" 291 src \u003d "\u003e

Let a \u003d 0, b \u003d h (n- height of a truncated cone), then https://pandia.ru/text/77/502/images/image030_16.gif "width \u003d" 36 "height \u003d" 17 src \u003d "\u003e \u003d. .

6. Bowl.

The ball can be obtained by rotating the circle with the center (0; 0) around the OX axis. The semicircle located above the OX axis is given by the equation

https://pandia.ru/text/77/502/images/image034_13.gif "width \u003d" 13 "height \u003d" 16 src \u003d "\u003e x r.

Besides finding an area of \u200b\u200ba flat figure with a specific integral (see 7.2.3.)the most important topic of the topic is calculation of the volume of rotation. The material is simple, but the reader must be prepared: you need to be able to solve uncertain integralsmiddle complexity and apply Newton-Leibnic formula in specific integral, ncould also also confident skills for building drawings. In general, in the integral calculus, many interesting applications, with the help of a specific integral, you can calculate the area of \u200b\u200bthe figure, the volume of the body of rotation, the length of the arc, the surface area of \u200b\u200bthe body and much more. Imagine some flat figure on the coordinate plane. Presented? ... Now this figure can also be rotated, and rotate in two ways:

- around the abscissa axis ;

- around the axis of the ordinate .

We will analyze both cases. The second way of rotation is particularly interesting, it causes the greatest difficulties, but in fact the decision is almost the same as in a more common rotation around the abscissa axis. Let's start with the most popular variety of rotation.

Calculation of body volume formed by rotation of a flat shape around the axis OX.

Example 1.

Calculate the volume of the body obtained by the rotation of the shape, limited lines, around the axis.

Decision:As in the task of finding the area, the decision begins with the drawing of a flat figure. That is, on the plane Xoy. It is necessary to build a figure limited by lines, and do not forget that the equation sets the axis. The drawing here is pretty simple:

The desired flat figure is shaded in a blue, it is she who rotates around the axis. As a result of rotation, such a slight egg-shaped flying plate with two sharp vertices on the axis OX., symmetrical about the axis OX.. In fact, the body has a mathematical name, look in the directory.

How to calculate the volume of the body of rotation? If the body is formed as a result of rotation around the axisOX.His mentally divided into parallel layers of small thickness dX.who are perpendicular to the axis OX.. The volume of the whole body is obviously the sum of the volume of such elementary layers. Each layer, like a round rode lemon, - a low cylinder height dX. and with the radius of the foundation f.(x.). Then the volume of one layer is the product of the base area π f. 2 to the height of the cylinder ( dX.), or π ∙ f. 2 (x.)∙dX.. And the area of \u200b\u200bthe entire body of rotation is the amount of elementary volumes, or the correspondingly determined integral. The volume of the body of rotation can be calculated by the formula:

How to place the limits of integration "A" and "BE", easy to guess from the drawing made. Function ... What is this function? Let's look at the drawing. Flat figure is limited to a parabola chart from above. This is the function that is meant in the formula. In practical tasks, a flat figure can sometimes be located below the axis OX.. It does not change anything - the function in the formula is erected into a square: f. 2 (x.), in this way, the volume of the rotation body is always non-negativethat is very logical. Calculate the scope of rotation using this formula:

As we have already noted, the integral is almost always simple, the main thing is to be attentive.

Answer:

In response, you must define the dimension - cubic units. That is, in our body of rotation approximately 3.35 "cubes". Why it is cubic units? Because it is the most universal wording. Cubic centimeters can be cubic meters, there may be cubic kilometers, etc., this is how many green men your imagination will be placed in a flying plate.

Example 2.

Find the volume of the body formed by rotation around the axis OX. Figures, limited lines ,,

This is an example for self-decide. Complete solution And the answer at the end of the lesson.

Example 3.

Calculate the volume of the body obtained during rotation around the abscissa axis of the shape, limited lines, and.

Decision:I will show a flat figure in the drawing, limited by lines ,,,,, not forgetting that equation x. \u003d 0 sets the axis Oy.:

The desired figure is shaded in blue. When it rotates around the axis OX. It turns out a flat angular bagel (washer with two conical surfaces).

The volume of the body of rotation is calculated as the difference in volumes. First consider the figure, which is circled in red. When it rotates around the axis OX. The truncated cone is obtained. Denote the volume of this truncated cone through V. 1 .

Consider a figure that is circled green. If you rotate this figure around the axis OX., I also will also have a truncated cone, just a little less. Denote its volume through V. 2 .

Obviously, the difference in volumes V. = V. 1 - V. 2 - this is the volume of our "bagel".

We use the standard formula for finding the volume of the body of rotation:

1) The figure circled in red is limited from above straight, so:

2) The figure visited green is limited from above straight, so:

3) the volume of the original body of rotation:

Answer:

It is curious that in this case the solution can be checked using the school formula to calculate the volume of a truncated cone.

The decision itself is more often arranged in short, approximately in such a spirit:

Definition 3. The body of rotation is the body obtained by rotating a flat figure around the axis that does not cross the figure and lying with it in the same plane.

The axis of rotation can and cross the figure, if it is the axis of the symmetry of the figure.

Theorem 2.
, axis
and straight cuts
and

rotates around the axis
. Then the volume of the resulting rotation body can be calculated by the formula

(2)

Evidence. For such a body, the cross section with the abscissa - this is a circle of radius
So
and formula (1) gives the required result.

If the figure is limited to the graphs of two continuous functions
and
, and straight cuts
and
Moreover
and
, when rotating around the abscissa axis, we obtain the body, the volume of which

Example 3. Calculate the volume of the torus obtained by the rotation of the circle limited by the circle

around the abscissa axis.

R measure. The specified circle bottom is limited by a graph
, and from above -
. The difference of squares of these functions:

The desired volume

(A graph of the integrand is the upper part-friendly, so the integral written above is the semicircular area).

Example 4. Parabolic segment
, and high , rotates around the base. Calculate the volume of the resulting body ("Lemon" Cavalieri).

R measure. Parabola is placed as shown in the figure. Then its equation
, and
. Find the value of the parameter :
. So, the desired volume:

Theorem 3. Let a curvilinear trapeze, limited by a chart of a continuous non-negative function
, axis
and straight cuts
and
Moreover
rotates around the axis
. Then the volume of the receiving body of rotation can be found by the formula

(3)

The idea of \u200b\u200bproof. Small cut
points

, part and spend direct
. The whole trapeze will decompose on strips, which can be considered approximately rectangles with the base.
and height
.

The cylinder is obtained when rotating such a rectangle, we will cut through the forming and unfold. We get "almost" parallelepiped with dimensions:
,
and
. Its volume
. So, for the volume of the body of rotation, we will have approximate equality

To obtain accurate equality, you need to go to the limit when
. The amount written above is the integral amount for the function
Therefore, in the limit, we obtain the integral from formula (3). Theorem is proved.

Note 1. In Theorems 2 and 3 Condition
you can omit: Formula (2) is generally insensitive to the sign
, and in formula (3) enough
replaced by
.

Example 5. Parabolic segment (base
, height ) Rates around the height. Find the volume of the resulting body.

Decision. Place a parabola as shown in the figure. And although the axis of rotation crosses the figure, it is the axis - is the axis of symmetry. Therefore, it is necessary to consider only the right half of the segment. Parabolla equation
, and
So
. We have for volume:

Note 2. If the curvilinear boundary of the curvilinear trapezium is set by parametric equations
,
,
and
,
you can use formulas (2) and (3) with replacement on the
and
on the
when it changes t. from
before .

Example 6. Figure is limited to the first arch cycloids
,
,
, and the abscissa axis. Find the volume of the body obtained by the rotation of this figure around: 1) the axis
; 2) axis
.

Decision. 1) general formula
In our case:

2) general formula
For our figure:

We offer students to independently carry out all the calculations.

Note 3. Let the curvilinear sector limited to the neur-rive
and rays
,

rotates around the polar axis. The volume of the resulting body can be calculated by the formula.

Example 7. Part of the shape limited cardioid
circumference
rotates around the polar axis. Find the volume of the body that it turns out.

Decision. Both lines, and therefore the figure that they limit is symmetrical with respect to the polar axis. Therefore, it is necessary to consider only the part for which
. Curves intersect
and

for
. Further, the figure can be considered as a difference of two sectors, which means the volume to calculate as the difference between the two integrals. We have:

Tasks For an independent solution.

1. Circular segment whose base
, height , rotates around the base. Find the scope of rotation.

2. Find the volume of the paraboloid of rotation, the base of which and the height is equal .

3. Figure limited by Astroide
,
rotates-smiling around the abscissa axis. Find the volume of the body that is obtained.

4. Figure limited lines
and
rates around the axis of the abscissa. Find the scope of rotation.

Type of lesson: Combined.

The purpose of the lesson: Learn to calculate the volume of rotation bodies using the integrals.

Tasks:

consolidate the ability to identify curvilinear trapecies from a number of geometric shapes and to work out the skill of calculating the area of \u200b\u200bcurvilinear trapezium;
get acquainted with the concept of a bulk figure;
learn to calculate the volume of the bodies of rotation;
promote the development of logical thinking, competent mathematical speech, accuracy in the construction of drawings;
brief interest in the subject, to the operating of mathematical concepts and images, to raise the will, independence, perseverance when the final result is reached.

During the classes

I. Organizational moment.

Greeting group. Message to students' targets.

Reflection. Calm melody.

- Today's lesson I would like to start with parables. "Sage lived, who knew everything. One person wanted to prove that the sage knows not all. Close in the palms of the butterfly, he asked: "Tell me, a sage, what a butterfly in my hands: Dead or Live?" And he thinks: "Says live - I will kill it, say dead - I will release." Sage, thinking, replied: "All in your hands". (Presentation.Slide)

- Therefore, let's work fruitfully today, we will acquire a new luggage of knowledge, and the resulting skills and skills will be applied in future life and practical activities. "All in your hands".

II. Repetition of the previously studied material.

- Let's remember the main points of the previously studied material. To do this, perform the task "Exclude the unnecessary word."(Slide.)

(The student goes to I.D. With the doctor removes the unnecessary word.)

- Right "Differential". Try the remaining words to call one common word. (Integral calculus.)

- Let's remember the main stages and concepts associated with integral calculus ..

"Mathematical cluster".

The task. Restore skip. (A student comes out and enters the handle the necessary words.)

- Abstract about the use of integrals, we heard later.

Work in notebooks.

- Newton's English physicist Newton (1643-1727) and German Philosopher Gottfried Leibnitsa (1646-1716) brought out the formula Newton labitsa (1643-1727). And this is not surprising, because mathematics is a language that nature speaks.

- Consider how this formula is used when solving practical tasks.

Example 1: Calculate the area of \u200b\u200bthe shape, limited lines

Solution: Build on the coordinate plane graphics of functions . We highlight the area of \u200b\u200bthe figure, which you need to find.

III. Studying a new material.

- Pay attention to the screen. What is shown in the first drawing? (Slide) (Flat figure is presented in the figure.)

- What is shown in the second drawing? Is this figure flat? (Slide) (The figure shows a bulk figure.)

- In space, on earth and in everyday life, we are found not only with flat figures, but also in bulk, but how to calculate the volume of such tel? For example, the volume of planets, fires, meteorite, etc.

- The volume of houses are conceived about the volume, and overflow water from one vessel to another. The rules and techniques for calculating volumes were to arise, another thing is how accurates they were and reasonable.

Message student. (Tyurina Vera.)

"Thus, the considered works of Kepler marked the beginning of a whole flow of studies that were crowned in the last quarter of the XVII century. Registration in the works of I. Newton and G.V. Differential and integral leibher. Mathematics of variables of magnitude occupied from this time the leading place in the system of mathematical knowledge.

- Today we are with you and will deal with such practical activities, therefore,

The theme of our lesson: "Calculation of rotation bodies using a specific integral". (Slide)

- You will find out the definition of the body of rotation by performing the following task.

"Labyrinth".

Labyrinth (Greek word) means a move in the dungeon. A labyrinth-tangled network of tracks, moves that communicate with each other.

But the definition "broke", there were tips in the form of arrows.

The task. Find the output from the confused position and write down the definition.

Slide. "Map instruction" calculation of volumes.

Using a specific integral, it is possible to calculate the volume of a body, in particular, the bodies of rotation.

The body of rotation is called the body obtained by the rotation of the curvilinear trapezium around its base (Fig. 1, 2)

The volume of the body of rotation is calculated by one of the formulas:

1. around the axis oh.

2. if the rotation of the curvilinear trapezium around the OU axis.

Card Instructing gets every student. The teacher emphasizes the highlights.

- The teacher explains the solution of examples on the board.

Consider an excerpt from famous fairy tales A. S. Pushkin "Tale of Tsar Saltan, about his son's nice and mighty Bogatyr Prince Gwidone Saltanovic and about the beautiful princess Swan" (Slide 4):

…..
And brought the messenger messenger
On the same day, the order is:
"The king makes his boyars,
Time not spending free
And queen and spodes
To secretly quit in the abyss of water. "
There is nothing to do: boyars,
Exhausted about the sovereign
And the queen is young
In the bedroom came to her crowd.
Announced Tsarisk Volodya -
She and his son a stake
Read out loud decree
And the queen in the same hour
In a barrel with his son, planted,
Scrambled, shown
And let them go to Okian -
So Tella de Tsar Saltan.

What should be the volume of the barrels so that the queen and her son fit in it?

- Consider the following tasks

1. Find the volume of the body obtained by rotation around the axis of the ordinate of a curvilinear trapezoid, limited lines: x 2 + y 2 \u003d 64, y \u003d -5, y \u003d 5, x \u003d 0.

Answer: 1163. cm. 3 .

Find the volume of the body obtained by the rotation of a parabolic trapezium around the abscissa axis y \u003d, x \u003d 4, y \u003d 0.

IV. Fastening a new material

Example 2. Calculate the volume of the body formed by the rotation of the petal around the abscissa axis y \u003d x 2, y 2 \u003d x.

We construct graphs of the function. y \u003d x 2, y 2 \u003d x. Schedule y 2 \u003d x We convert to the form y.= .

Have V \u003d V 1 - V 2 Calculate the volume of each function

"Now, let's consider the tower for a radio station in Moscow on Shabolovka, built on the project of a wonderful Russian engineer, Honorary Academician V. G. Shukhov. It consists of parts - rotation hyperboloids. Moreover, each of them is made of straight metal rods connecting the adjacent circle (Fig. 8, 9).

- Consider the task.

Find the volume of the body obtained by the rotation of the arc hyperboles Around her imaginary axis, as shown in Fig. 8, where

cube units.

Tasks for groups. Students pull out lots with tasks, drawings are performed on Watman, one of the representatives of the group protects work.

1st group.

Hit! Hit! Still a blow!
Flies in the goal the ball - the ball!
And this is the ball watermelon
Green, round, tasty.
Visit better - ball What!
It is made of some circles.
Cut on the circles of watermelon
And try to taste them.

Find the volume of the body obtained by rotation around the axis oh function limited

Error! The bookmark is not defined.

- Tell me, please, where do we meet with this figure?

House. Task for 1 group. CYLINDER (slide) .

"Cylinder - what is?" I asked my dad.
Father laughed: the cylinder is a hat.
To have a true presentation
Cylinder, let's say it is canning bank.
Parot Pipe - Cylinder,
Pipe on our roof - too,

All pipes on the cylinder are similar.
And I brought an example of this -
Kaleidoscope my favorite,
Eye from him you will not turn around,
And also the cylinder is like.

- The task. HOME MACHINE Create a graph of the function and calculate the volume.

2nd group. CONE (slide).

Mom said: And now
About the cone there will be my story.
In a high hat, starvature
He believes the stars all year round.
Cone - Star Hat.
That's what he is. Understood? That's something.
Moma has a table,
Bottles spilled oil.
- Where is the funnel? There are no funnels.
Look for. Do not stand on the sidelines.
- Mom, I will not be touched from the spot,
Tell me about the cone.
- The funnel is in the form of a cone leak.
Well, find me quarrel.
I could not find a funnel,
But Mom made a kulok,
Cardboard has enveloped
And cleverly fastened.
Oil pour, Mom is glad,
Cone came out what it is necessary.

The task. Calculate the volume of the body obtained by rotation around the abscissa axis

House. Task for the 2nd group. PYRAMID (slide).

I saw a picture. In this picture
There is a pyramid in the sandy desert.
Everything in the pyramid is unusually
Some kind of riddle and mystery in it.
And the Spasskaya Tower on the square of the Red
Both children and adults are familiar perfectly.
You look at the tower - the usual look,
And what on top of it? Pyramid!

The task. Home work Create a graph of the function and calculate the volume of the pyramid

- Volumes of different bodies, we calculated based on the basic formula of body volumes using the integral.

This is another confirmation that a certain integral is some foundation for learning mathematics.

- Well, now let's take a little rest.

Find a couple.

Mathematical domino tune plays.

"The way that he searched himself, will not be overwhelmed ..."

Research. Application integral in economics and technology.

Tests for strong students and mathematical football.

Mathematical simulator.

2. The combination of all the primary from this function is called

A) uncertain integral

B) function,

C) differentiation.

7. Find the volume of the body obtained by rotation around the abscissa axis of the curvilinear trapezium limited to the lines:

D / s. Calculate the volume of the bodies of rotation.

Reflection.

Reception of reflection in the form sincweune. (fivesty).

The 1st line is the name of the topic (one noun).

The 2nd line is a description of the topic in two words, two adjectives.

3rd line - description of the action within the framework of this topic in three words.

The 4th line is the phrase of their four words, shows the relation to the topic (a whole sentence).

The 5th line is synonym that repeats the essence of the theme.

Volume.
Certain integral, integrable function.
We build, rotate, calculate.
The body obtained by the rotation of the curvilinear trapezium (around its base).
The body of rotation (volumetric geometric body).

Output (slide).

A certain integral is some foundation for learning mathematics, which makes an indispensable contribution to solving the tasks of practical content.
The topic "Integral" brightly demonstrates the connection of mathematics with physics, biology, economy and technology.
The development of modern science is unthinkable without using the integral. In this regard, it is necessary to start studying in the framework of secondary special education!

Estimation. (Commenting.)

Great Omar Khayam - mathematician, poet, philosopher. He calls to be masters of his fate. We listen to the excerpt from his work:

You will say, this life is one moment.
Her appreciation, in her draws inspiration.
How to spend it and pass.
Do not forget: she is your creation.