Figure Limited Lines Find the volume of the body of rotation. Integral in business

How to calculate the scope of rotation
Using a specific integral?

In general, in integral calculus, there are a lot of interesting applications, with the help of a specific integral, the size of the figure, the volume of the body of rotation, the length of the arc, the surface area of \u200b\u200bthe rotation and much more are calculated. Therefore, it will be fun, please set up to optimistic way!

Imagine some flat figure on the coordinate plane. Presented? ... I wonder who I presented ... \u003d))) We have already found it. But, in addition, this figure can also be rotated, and rotate in two ways:

- around the abscissa axis;
- Around the ordinate axis.

This article will disassemble 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. As a bonus, I will return to the task of finding the area of \u200b\u200bthe figureAnd I will tell you how to find the area in the second way - along the axis. Not even so much bonus, how much material successfully fits into the subject.

Let's start with the most popular variety of rotation.

flat shape around the axis

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 square, the decision begins with the drawing of a flat figure. That is, on the plane it is necessary to build a figure limited by lines, and not forget that the equation sets the axis. How rational and faster perform the drawing, you can learn on the pages Charts and properties of elementary functionsand. This is a Chinese reminder, and on this moment I no longer stop.

The drawing here is pretty simple:

The desired flat figure is shaded in blue, it is she who rotates around the axis as a result of rotation, it turns out such a slight egg-shaped flying plate, which is symmetrical with respect to the axis. In fact, the body has a mathematical name, but according to the directories something too lazy to check, so we are going on.

How to calculate the volume of the body of rotation?

The volume of the rotation body can be calculated by the formula:

In the formula, the integral is necessarily present. It was so necessary - everything that spins in life is associated with this constant.

How to arrange the limits of integration "A" and "BE", I think it is 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 Flat figure can sometimes be located below the axis. It does not change anything - the integrated function in the formula is erected into a square: thus the integral is always nonnegatorythat is very logical.

Calculate the scope of rotation using this formula:

As I 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 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.

Find body volume educated by rotation around the axis shape, limited lines ,,

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

Consider two more complex taskswho are also often found in practice.

Calculate the volume of the body obtained when rotating around the abscissa axis of the figure limited lines, and

Decision: Show a flat figure, limited by lines ,,,,,,,, the equation sets the axis:

The desired figure is shaded in blue. When it rotates around the axis, such a surreal bagel with four angles is obtained.

The volume of the body of rotation is calculated as the difference in volumes.

First consider the figure, which is circled in red. With its rotation around the axis, a truncated cone is obtained. Denote the volume of this truncated cone through.

Consider a figure that is circled green. If you rotate this figure around the axis, you will also get a truncated cone, only a little smaller. Denote its volume through.

And, obviously, the difference in volumes is exactly 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:

Now a little rest, and tell about the geometric illusions.

People often have illusions associated with the volume, which was noticed by Perelman (another) in the book Entertaining geometry. Look at the flat figure in the tried task - it seems to be small in the area, and the volume of the body of rotation is slightly over 50 cubic units, which seems too much. By the way, the average man in his entire life drinks liquid with a room with an area of \u200b\u200b18 square metersthat, on the contrary, it seems too small.

After lyric retreat, it is relevant to solve the creative task:

Calculate the volume of the body formed by the rotation relative to the axis of the flat figure, limited by lines, where.

This is an example for an independent solution. Please note that all matters occur in the strip, in other words, the finished integration limits are actually given. Properly draw graphs trigonometric functions, I remind the material of the lesson about geometric chart transformations: If the argument is divided into two: then the graphs are stretched along the axis twice. It is advisable to find at least 3-4 points according to trigonometric tablesIn order to more accurately perform the drawing. Complete solution and answer at the end of the lesson. By the way, the task can be solved rationally and is not very rational.

Calculating the volume of the body formed by rotation
flat shape around the axis

The second paragraph will be even more interesting than the first. The task to calculate the volume of the body of rotation around the axis of the ordinate is also a fairly frequent guest in test work. Along the way will be considered the task of finding the area of \u200b\u200bthe figure In the second way - integrating on the axis, this will allow you not only to improve your skills, but also teach the most favorable way to solve. There is practical life meaning in it! As a smile, my teacher remembered on the method of teaching mathematics, many graduates thanked her words: "We really helped your subject, now we are effective managers and optimally led by personnel." Taking this opportunity, I also express her great gratitude, especially since I use the knowledge gained direct destination =).

I recommend to read all, even full teapots. Moreover, the learned material of the second paragraph will have invaluable assistance in calculating double integrals..

Dana Flat figure, limited lines , , .

1) Find the area of \u200b\u200ba flat figure limited by these lines.
2) Find the volume of the body obtained by the rotation of a flat figure limited by these lines around the axis.

Attention! Even if you want to familiarize yourself with the second item, first read the first one!

Decision: The task consists of two parts. Let's start with the square.

1) Perform drawing:

It is easy to see that the function sets the upper branch of the parabola, and the function is the lower branch of the parabola. Before us is a trivial parabola that "lies on the side."

The desired figure, the area of \u200b\u200bwhich is to be found, shaded in blue.

How to find the area of \u200b\u200bthe figure? It can be found "ordinary" way, which was considered at the lesson Certain integral. How to calculate the area of \u200b\u200bthe figure. Moreover, the area of \u200b\u200bthe figure is like the amount of the area:
- On the cut ;
- On the segment.

Therefore:

What is in this case the usual way to solve? First, it turned out two integrals. Secondly, under the integrals of the roots, and the roots in the integrals are not a gift, besides, you can get confused in the substitution of the integration limits. In fact, the integrals, of course, are not murder, but in practice everything is much sadder, I just picked up for the task of the "better" function.

There is a more rational solution path: it consists in the transition to reverse functions and integration along the axis.

How to go to reverse functions? Roughly speaking, you need to express "X" through "Irek". First we will deal with parabola:

This is enough, but make sure that the same function can be removed from the bottom branch:

With straight, everything is easier:

Now we look at the axis: Please, periodically tilt your head to the right of 90 degrees along the course of the explanation (this is not a joke!). The figure we need lies on the segment, which is marked with a red dotted line. At the same time, on the segment, the direct is located above parabola, and therefore the area of \u200b\u200bthe figure should be found on the formula already familiar to you: . What changed in the formula? Only the letter, and nothing more.

! Note: Limits of integration over the axis should be placed strictly bottom up!

Find area:

On the segment, so:

Please note how I implemented integration is the most rational way, and in the next point of the task will be clear - why.

For readers who doubt integration correctness, I will find derivatives:

The initial integrand is obtained, it means that the integration is made correctly.

Answer:

2) Calculate the volume of the body formed by the rotation of this figure around the axis.

Redrawing drawing a little in another design:

So, the figure, shaded in blue, rotates around the axis. As a result, it turns out a "hung butterfly" that spins around its axis.

To find the volume of the body of rotation, we will integrate along the axis. First you need to go to reverse functions. This is already done and described in detail in the previous paragraph.

Now we til off the right again and we study our figure. Obviously, the volume of the body of rotation should be found as a difference in volumes.

Rotate the figure, circled in red, around the axis, resulting in a truncated cone. Denote this volume through.

Rotate the figure, circled with green, around the axis and denote through the volume of the body of rotation.

The volume of our butterfly is equal to the difference in volumes.

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

What is the difference from the formula of the previous paragraph? Only in the letter.

But the advantage of integration, which I recently spoke is much easier to find than to pre-build a replacement function in the 4th degree.

Answer:

Note that if the same flat figure rotate around the axis, then it will turn out a completely different body of rotation, of the other, naturally, the volume.

Dana flat figure limited lines and axes.

1) Go to reverse functions and find the area of \u200b\u200ba flat figure limited by these lines, integrating via variable.
2) Calculate the volume of the body obtained by rotating a flat figure limited by these lines around the axis.

This is an example for an independent solution. Those who wish can also find the figure of the figure "ordinary" way, thereby performing the check of paragraph 1). But if, I repeat, you will rotate a flat figure around the axis, then you will get a completely different body of rotation with another volume, by the way, the correct answer (also for lovers to blend).

The full solution of the two proposed task items at the end of the lesson.

Yes, and do not forget to tilt the head to the right to figure out the bodies of rotation and within the integration!

Wanted it was already, finish the article, but today they brought interesting example Just to find the volume of the body of rotation around the ordinate axis. Flushing:

Calculate the volume of the body formed by rotation around the axis of the figure, limited by curves and.

Decision: Perform drawing:

Along the way I get acquainted with the graphs of some other functions. Such a interesting schedule even function ….

How to calculate the scope of rotation using a specific integral?

In addition to finding a flat figure with a specific integral 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 integrals Middle complexity and apply Newton-Leibnic formula in certain integral . As for the task of finding the area, you need confident skills to build drawings - this is almost the most important (since the integrals themselves will be easily lightly). You can master the competent and fast technique for constructing graphs using a methodical material. . But, in fact, about the importance of the drawings, I repeatedly spoke at the lesson .

In general, in integral calculus, there are a lot of interesting applications, with the help of a specific integral, the area of \u200b\u200bthe figure, the volume of the body, the length of the arc, the body surface area and much more are possible. Therefore, it will be fun, please set up to optimistic way!

– around the abscissa axis; - Around the ordinate axis.

This article will disassemble 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. As a bonus, I will return to the task of finding the area of \u200b\u200bthe figure And I will tell you how to find the area in the second way - along the axis. Not even so much bonus, how much material successfully fits into the subject.

Let's start with the most popular variety of rotation.

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 it is necessary to build a figure limited by lines, while do not forget that the equation is the axis. How rational and faster perform the drawing, you can learn on the pages Charts and properties of elementary functions and Certain integral. How to calculate the area of \u200b\u200bthe figure . This is a Chinese reminder, and at this moment I no longer stop.

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, which is symmetrical relative to the axis. In fact, the body has a mathematical name, but in the reference book something too lazy to look, so we are going on.

How to calculate the volume of the body of rotation?

The volume of the body of rotation can be calculated by the formula:

In the formula, the integral is necessarily present. It was so necessary - everything that spins in life is associated with this constant.

How to arrange the limits of integration "A" and "BE", I think it is easy to guess from the drawing made.

Function ... What is this function? Let's look at the drawing. Flat figure is limited to the Parabolys top schedule. This is the function that is meant in the formula.

In practical tasks, a flat figure can sometimes be located below the axis. It does not change anything - the function in the formula is built into the square: so the volume of the rotation body is always non-negativethat is very logical.

Calculate the scope of rotation using this formula:

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

Answer:

Example 2.

Find the volume of the body formed by the rotation around the shape axis limited by lines ,,

This is an example for an independent solution. Complete solution and answer at the end of the lesson.

Consider two more complex tasks that are also common in practice.

Example 3.

Calculate the volume of the body obtained when rotating around the abscissa axis of the figure limited lines ,, and

Decision:I will show a flat figure in the drawing, limited by lines ,,,, not forgetting that equation is the axis:

The desired figure is shaded in blue. When it rotates around the axis, such a surreal bagel with four angles is obtained.

The volume of the body of rotation is calculated as the difference in volumes.

First consider the figure, which is circled in red. With its rotation around the axis, a truncated cone is obtained. Denote the volume of this truncated cone through.

Consider a figure that is circled with green. If you rotate this figure around the axis, you will also get a truncated cone, only a little smaller. Denote its volume through.

And, obviously, the difference in volumes is exactly 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:

Now a little rest, and tell about the geometric illusions.

People often have illusions associated with the volume, which was noticed by Perelman (not) in the book Entertaining geometry. Look at the flat figure in the tried task - it seems to be small in the area, and the volume of the body of rotation is slightly over 50 cubic units, which seems too much. By the way, the average person in his entire life drinks a liquid with a room with an area of \u200b\u200b18 square meters, which, on the contrary, seems too small.

In general, the education system in the USSR was really the best. The same Book of Perelman, written in 1950, develops very well, as a humorist said, converted and teaches to look for original non-standard solutions to problems. Recently, some chapters read with great interest, recommend, accessible even for humanitarian. No, you do not need to smile that I offered an impact pastime, erudition and a wide range of communication - a great thing.

After lyric retreat, it is relevant to solve the creative task:

Example 4.

Calculate the volume of the body formed by the rotation relative to the axis of the flat figure limited by the lines, where.

This is an example for an independent solution. Please note that all matters occur in the strip, in other words, almost ready-made integration limits are given. Also try to correctly draw graphs of trigonometric functions, if the argument is divided into two:, then the graphs are stretched by ledging twice. Try to find at least 3-4 points according to trigonometric tables And more precisely perform the drawing. Complete solution and answer at the end of the lesson. By the way, the task can be solved rationally and is not very rational.

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

The second paragraph will be even more interesting than the first. The task for calculating the volume of the body of rotation around the axis of the ordinate is also a fairly frequent guest in the tests. Along the way will be considered the task of finding the area of \u200b\u200bthe figure In the second way - integrating on the axis, this will allow you not only to improve your skills, but also teach the most favorable way to solve. There is practical life meaning in it! As a smile, my teacher remembered on the method of teaching mathematics, many graduates thanked her words: "We really helped your subject, now we are effective managers and optimally led by personnel." Taking this opportunity, I also express her my great gratitude, especially since I use the knowledge gained on direct purpose \u003d).

Example 5.

Dana flat figure limited by lines ,,.

1) Find the area of \u200b\u200ba flat figure limited by these lines. 2) Find the volume of the body obtained by the rotation of a flat figure limited by these lines around the axis.

Attention! Even if you want to get acquainted only with the second item, first before Read the first!

Decision: The task consists of two parts. Let's start with the square.

1) Perform drawing:

It is easy to notice that the function sets the upper branch of the parabola, and the function is the bottom branch of the parabola. Before us is a trivial parabola that "lies on the side."

The desired figure, the area of \u200b\u200bwhich is to be found, shaded in blue.

How to find the area of \u200b\u200bthe figure? It can be found "ordinary" way, which was considered at the lesson Certain integral. How to calculate the area of \u200b\u200bthe figure . Moreover, the area of \u200b\u200bthe figure is located as the amount of the area: - on the segment ; - On the segment.

Therefore:

There is a more rational solution path: it consists in the transition to reverse functions and integration along the axis.

How to go to reverse functions? Roughly speaking, you need to express "X" through "Irek". First we will deal with parabola:

This is enough, but make sure that the same function can be removed from the bottom branch:

With straight, everything is easier:

Now we look at the axis: Please, periodically tilt your head to the right of 90 degrees along the course of the explanation (this is not a joke!). The figure we need lies on the segment, which is marked with a red dotted line. At the same time, the parabolava, and therefore the area of \u200b\u200bthe figure should be found on the formula already familiar to you: . What changed in the formula? Only the letter, and nothing more.

! Note: The axis integration limits should be arranged.strictly bottom up !

Find area:

On the segment, so:

Please note how I implemented integration is the most rational way, and in the next point of the task will be clear - why.

For readers who doubt integration correctness, I will find derivatives:

The initial integrand is obtained, it means that the integration is made correctly.

Answer:

2) Calculate the volume of the body formed by the rotation of this figure around the axis.

Redrawing drawing a little in another design:

So, the figure, shaded in blue, rotates around the axis. As a result, it turns out a "hung butterfly" that spins around its axis.

To find the volume of the body of rotation, we will integrate along the axis. First you need to go to reverse functions. This is already done and described in detail in the previous paragraph.

Now we til off the right again and we study our figure. Obviously, the volume of the body of rotation should be found as a difference in volumes.

Rotate the figure, circled in red, around the axis, resulting in a truncated cone. Denote this volume through.

Rotate the figure, circled with green, around the axis and indicate the volume of the resulting body of rotation.

The volume of our butterfly is equal to the difference in volumes.

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

What is the difference from the formula of the previous paragraph? Only in the letter.

But the advantage of integration, which I recently spoke is much easier to find than to pre-build a replacement function in the 4th degree.

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 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.

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. (The student comes out and enters the handle 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 meet 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.)

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. (Slide 2)

"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 practical activities, hence,

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

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. 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.
Development modern science 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.

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 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.

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. (A 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 this formula is used when solving practical tasks.

Example 1: Calculate the area of \u200b\u200bthe shape, 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 are found not only with flat figures, but also volume, 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.

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, therefore,

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

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. Fastening a new material

Calculate the volume of the body formed by the rotation of the 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.

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!

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.

The volume of the body of rotation can be calculated by the formula:

In the formula, the integral is necessarily present. It was so necessary - everything that spins in life is associated with this constant.

How to arrange the limits of integration "A" and "BE", I think it is 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. It does not change anything - the function in the formula is erected into a square: thus the volume of the rotation body is always non-negativethat is very logical.

Calculate the scope of rotation using this formula:

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

Answer:

Example 2.

Find the volume of the body formed by rotation around the shape axis limited lines ,,

This is an example for an independent solution. Complete solution and answer at the end of the lesson.

Consider two more complex tasks that are also common in practice.

Example 3.

Calculate the volume of the body obtained when rotating around the abscissa axis of the figure limited lines, and

Decision:Showing a flat figure, limited by lines, ,,, not forgetting that the equation sets the axis: