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The volume of a body formed by rotation around an online axis. Calculation of volumes of bodies of revolution using a definite integral

Topic: “Calculating the volumes of bodies of revolution using definite integral»

Lesson type: combined.

The purpose of the lesson: learn to calculate the volumes of bodies of revolution using integrals.

Tasks:

consolidate the ability to identify curved trapezoids from a series geometric shapes and practice the skill of calculating the areas of curvilinear trapezoids;

get acquainted with the concept of a three-dimensional figure;

learn to calculate the volumes of bodies of revolution;

promote development logical thinking, competent mathematical speech, accuracy when constructing drawings;

to cultivate interest in the subject, in operating with mathematical concepts and images, to cultivate will, independence, and perseverance in achieving the final result.

During the classes

I. Organizational moment.

Greetings from the group. Communicate lesson objectives to students.

I would like to start today's lesson with a parable. “Once upon a time there lived a wise man who knew everything. One man wanted to prove that the sage does not know everything. Holding a butterfly in his hands, he asked: “Tell me, sage, which butterfly is in my hands: dead or alive?” And he thinks: “If the living one says, I’ll kill her; if the dead one says, I’ll release her.” The sage, after thinking, replied: “Everything is in your hands.”

Therefore, let’s work fruitfully today, acquire a new store of knowledge, and we will apply the acquired skills and abilities in future life and in practical activities. “Everything is in your hands.”

II. Repetition of previously studied material.

Let's remember the main points of the previously studied material. To do this, let’s complete the task “Exclude superfluous word”.

(Students say an extra word.)

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

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

Exercise. Recover the gaps. (The student comes out and writes in the required words with a marker.)

Work in notebooks.

The Newton-Leibniz formula was derived by the English physicist Isaac Newton (1643-1727) and the German philosopher Gottfried Leibniz (1646-1716). And this is not surprising, because mathematics is the language spoken by nature itself.

Let's consider how this formula is used to solve practical problems.

Example 1: Calculate the area of a figure bounded by lines

Solution: Let's construct graphs of functions on the coordinate plane . Let's select the area of the figure that needs to be found.

III. Learning new material.

Pay attention to the screen. What is shown in the first picture? (The figure shows a flat figure.)

What is shown in the second picture? Is this figure flat? (The figure shows a three-dimensional figure.)

In space, on earth and in Everyday life We encounter not only flat figures, but also three-dimensional ones, but how can we calculate the volume of such bodies? For example: the volume of a planet, comet, meteorite, etc.

People think about volume both when building houses and when pouring water from one vessel to another. Rules and techniques for calculating volumes had to emerge; how accurate and justified they were is another matter.

The year 1612 was very fruitful for the residents of the Austrian city of Linz, where the famous astronomer Johannes Kepler lived, especially for grapes. People were preparing wine barrels and wanted to know how to practically determine their volumes.

Thus, the considered works of Kepler marked the beginning of a whole stream of research that culminated in the last quarter of the 17th century. design in the works of I. Newton and G.V. Leibniz of differential and integral calculus. From that time on, the mathematics of variables took a leading place in the system of mathematical knowledge.

Today we will do this practical activities, hence,

The topic of our lesson: “Calculating the volumes of bodies of rotation using a definite integral.”

You will learn the definition of a body of revolution by completing the following task.

“Labyrinth”.

Exercise. Find a way out of the confusing situation and write down the definition.

IVCalculation of volumes.

Using a definite integral, you can calculate the volume of a particular body, in particular, a body of revolution.

A body of revolution is a body obtained by rotation curved trapezoid around its base (Fig. 1, 2)

The volume of a body of revolution is calculated using one of the formulas:

1. around the OX axis.

2. , if the rotation of a curved trapezoid around the axis of the op-amp.

Students write down basic formulas in a notebook.

The teacher explains the solutions to the examples on the board.

1. Find the volume of the body obtained by rotating around the ordinate axis of a curvilinear trapezoid bounded by lines: x2 + y2 = 64, y = -5, y = 5, x = 0.

Solution.

Answer: 1163 cm3.

2. Find the volume of the body obtained by rotating a parabolic trapezoid around the x-axis y = , x = 4, y = 0.

Solution.

V. Math simulator.

2. The set of all antiderivatives of a given function is called

A) indefinite integral,

B) function,

B) differentiation.

7. Find the volume of the body obtained by rotating around the abscissa axis of a curvilinear trapezoid bounded by lines:

D/Z. Consolidating new material

Calculate the volume of the body, formed by rotation petal, around the x-axis y = x2, y2 = x.

Let's build graphs of the function. y = x2, y2 = x. Let's transform the graph y2 = x to the form y = .

We have V = V1 - V2 Let’s calculate the volume of each function:

Conclusion:

The definite integral is a certain foundation for the study of mathematics, which makes an irreplaceable contribution to solving practical problems.

The topic “Integral” clearly demonstrates the connection between mathematics and physics, biology, economics and technology.

Development modern science is unthinkable without using the integral. In this regard, it is necessary to begin studying it within the framework of secondary specialized education!

VI. Grading.(With commentary.)

The great Omar Khayyam - mathematician, poet, philosopher. He encourages us to be masters of our own destiny. Let's listen to an excerpt from his work:

You say, this life is one moment.
Appreciate it, draw inspiration from it.
As you spend it, so it will pass.
Don't forget: she is your creation.

Lesson type: combined.

The purpose of the lesson: learn to calculate the volumes of bodies of revolution using integrals.

Tasks:

consolidate the ability to identify curvilinear trapezoids from a number of geometric figures and develop the skill of calculating the areas of curvilinear trapezoids;
get acquainted with the concept of a three-dimensional figure;
learn to calculate the volumes of bodies of revolution;
promote the development of logical thinking, competent mathematical speech, accuracy when constructing drawings;
to cultivate interest in the subject, in operating with mathematical concepts and images, to cultivate will, independence, and perseverance in achieving the final result.

During the classes

I. Organizational moment.

Greetings from the group. Communicate lesson objectives to students.

Reflection. Calm melody.

– I would like to start today’s lesson with a parable. “Once upon a time there lived a wise man who knew everything. One man wanted to prove that the sage does not know everything. Holding a butterfly in his hands, he asked: “Tell me, sage, which butterfly is in my hands: dead or alive?” And he himself thinks: “If the living one says, I will kill her; the dead one will say, I will release her.” The sage, after thinking, replied: "All in your hands". (Presentation.Slide)

– Therefore, let’s work fruitfully today, acquire a new store of knowledge, and we will apply the acquired skills and abilities in future life and in practical activities. "All in your hands".

II. Repetition of previously studied material.

– Let’s remember the main points of the previously studied material. To do this, let's complete the task “Eliminate the extra word.”(Slide.)

(The student goes to I.D. uses an eraser to remove the extra word.)

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

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

“Mathematical bunch”.

Exercise. Recover the gaps. (The student comes out and writes in the required words with a pen.)

– We will hear an abstract on the application of integrals later.

Work in notebooks.

– The Newton-Leibniz formula was derived by the English physicist Isaac Newton (1643–1727) and the German philosopher Gottfried Leibniz (1646–1716). And this is not surprising, because mathematics is the language spoken by nature itself.

– Let’s consider how this formula is used to solve practical problems.

Example 1: Calculate the area of a figure bounded by lines

Solution: Let's build graphs of functions on the coordinate plane . Let's select the area of the figure that needs to be found.

III. Learning new material.

– Pay attention to the screen. What is shown in the first picture? (Slide) (The figure shows a flat figure.)

– What is shown in the second picture? Is this figure flat? (Slide) (The figure shows a three-dimensional figure.)

– In space, on earth and in everyday life, we encounter not only flat figures, but also three-dimensional ones, but how can we calculate the volume of such bodies? For example, the volume of a planet, comet, meteorite, etc.

– People think about volume both when building houses and when pouring water from one vessel to another. Rules and techniques for calculating volumes had to emerge; how accurate and reasonable they were is another matter.

Message from a student. (Tyurina Vera.)

– Thus, the considered works of Kepler laid the foundation for a whole stream of research that culminated in the last quarter of the 17th century. design in the works of I. Newton and G.V. Leibniz of differential and integral calculus. From that time on, the mathematics of variables took a leading place in the system of mathematical knowledge.

– Today you and I will engage in such practical activities, therefore,

The topic of our lesson: “Calculating the volumes of bodies of rotation using a definite integral.” (Slide)

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

“Labyrinth”.

Labyrinth (Greek word) means going underground. A labyrinth is an intricate network of paths, passages, and interconnecting rooms.

But the definition was “broken,” leaving clues in the form of arrows.

Exercise. Find a way out of the confusing situation and write down the definition.

Slide. “Map instruction” Calculation of volumes.

Using a definite integral, you can calculate the volume of a particular body, in particular, a body of revolution.

A body of revolution is a body obtained by rotating a curved trapezoid around its base (Fig. 1, 2)

The volume of a body of rotation is calculated using one of the formulas:

1. around the OX axis.

2. , if the rotation of a curved trapezoid around the axis of the op-amp.

Each student receives an instruction card. The teacher emphasizes the main points.

– The teacher explains the solutions to the examples on the board.

Let's consider an excerpt from the famous fairy tale by A. S. Pushkin “The Tale of Tsar Saltan, of his glorious and mighty son Prince Guidon Saltanovich and of the beautiful Princess Swan” (Slide 4):

…..
And the drunken messenger brought
On the same day the order is as follows:
“The king orders his boyars,
Without wasting time,
And the queen and the offspring
Secretly throw into the abyss of water.”
There is nothing to do: boyars,
Worrying about the sovereign
And to the young queen,
A crowd came to her bedroom.
They declared the king's will -
She and her son have an evil share,
We read the decree aloud,
And the queen at the same hour
They put me in a barrel with my son,
They tarred and drove away
And they let me into the okiyan -
This is what Tsar Saltan ordered.

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

– Consider the following tasks

1. Find the volume of the body obtained by rotating around the ordinate axis of a curvilinear trapezoid bounded by lines: x 2 + y 2 = 64, y = -5, y = 5, x = 0.

Answer: 1163 cm 3 .

Find the volume of the body obtained by rotating a parabolic trapezoid around the abscissa axis y = , x = 4, y = 0.

IV. Consolidating new material

Example 2. Calculate the volume of the body formed by the rotation of the petal around the x-axis y = x 2 , y 2 = x.

Let's build graphs of the function. y = x 2 , y 2 = x. Schedule y2 = x convert to the form y= .

We have V = V 1 – V 2 Let's calculate the volume of each function

– Now, let’s look at the tower for the radio station in Moscow on Shabolovka, built according to the design of the remarkable Russian engineer, honorary academician V. G. Shukhov. It consists of parts - hyperboloids of rotation. Moreover, each of them is made of straight metal rods connecting adjacent circles (Fig. 8, 9).

- Let's consider the problem.

Find the volume of the body obtained by rotating the hyperbola arcs around its imaginary axis, as shown in Fig. 8, where

cube units

Group assignments. Students draw lots with tasks, draw drawings on whatman paper, and one of the group representatives defends the work.

1st group.

Hit! Hit! Another blow!
The ball flies into the goal - BALL!
And this is a watermelon ball
Green, round, tasty.
Take a better look - what a ball!
It is made of nothing but circles.
Cut the watermelon into circles
And taste them.

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

Error! The bookmark is not defined.

– Please tell me where we meet this figure?

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

"Cylinder - what is it?" – I asked my dad.
The father laughed: The top hat is a hat.
To have a correct idea,
A cylinder, let's say, is a tin can.
Steamboat pipe - cylinder,
The pipe on our roof too,

All pipes are similar to a cylinder.
And I gave an example like this -
Kaleidoscope My love,
You can't take your eyes off him,
And it also looks like a cylinder.

- Exercise. Homework graph the function and calculate the volume.

2nd group. CONE (slide).

Mom said: And now
My story will be about the cone.
Stargazer in a high hat
Counts the stars all year round.
CONE - stargazer's hat.
That's what he is like. Understood? That's it.
Mom was standing at the table,
I poured oil into bottles.
-Where is the funnel? No funnel.
Look for it. Don't stand on the sidelines.
- Mom, I won’t budge.
Tell us more about the cone.
– The funnel is in the form of a watering can cone.
Come on, find her for me quickly.
I couldn't find the funnel
But mom made a bag,
I wrapped the cardboard around my finger
And she deftly secured it with a paper clip.
The oil is flowing, mom is happy,
The cone came out just right.

Exercise. Calculate the volume of a body obtained by rotating around the abscissa axis

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

I saw the picture. In this picture
There is a PYRAMID in the sandy desert.
Everything in the pyramid is extraordinary,
There is some kind of mystery and mystery in it.
And the Spasskaya Tower on Red Square
It is very familiar to both children and adults.
If you look at the tower, it looks ordinary,
What's on top of it? Pyramid!

Exercise. Homework: graph the function and calculate the volume of the pyramid

– We calculated the volumes of various bodies based on the basic formula for the volumes of bodies using an integral.

This is another confirmation that the definite integral is some foundation for the study of mathematics.

- Well, now let's rest a little.

Find a pair.

Mathematical domino melody plays.

“The road that I myself was looking for will never be forgotten...”

Research work. Application of the integral in economics and technology.

Tests for strong students and mathematical football.

Math simulator.

2. The set of all antiderivatives of a given function is called

A) an indefinite integral,

B) function,

B) differentiation.

7. Find the volume of the body obtained by rotating around the abscissa axis of a curvilinear trapezoid bounded by lines:

D/Z. Calculate the volumes of bodies of rotation.

Reflection.

Reception of reflection in the form syncwine(five lines).

1st line – topic name (one noun).

2nd line – description of the topic in two words, two adjectives.

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

The 4th line is a phrase of four words that shows the attitude to the topic (a whole sentence).

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

Volume.
Definite integral, integrable function.
We build, we rotate, we calculate.
A body obtained by rotating a curved trapezoid (around its base).
Body of rotation (volumetric geometric body).

Conclusion (slide).

A definite integral is a certain foundation for the study of mathematics, which makes an irreplaceable contribution to solving practical problems.
The topic “Integral” clearly demonstrates the connection between mathematics and physics, biology, economics and technology.
The development of modern science is unthinkable without the use of the integral. In this regard, it is necessary to begin studying it within the framework of secondary specialized education!

Grading. (With commentary.)

The great Omar Khayyam - mathematician, poet, philosopher. He encourages us to be masters of our own destiny. Let's listen to an excerpt from his work:

You will say, this life is one moment.
Appreciate it, draw inspiration from it.
As you spend it, so it will pass.
Don't forget: she is your creation.

flat figure around an axis

Example 3

Given a flat figure bounded by lines , , .

1) Find the area of a flat figure bounded by these lines.

2) Find the volume of the body obtained by rotating a flat figure bounded by these lines around the axis.

Attention! Even if you only want to read the second point, first Necessarily read the first one!

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

1) Let's make a drawing:

It is easy to see that the function specifies the upper branch of the parabola, and the function specifies the lower branch of the parabola. Before us is a trivial parabola that “lies on its side.”

The desired figure, the area of which is to be found, is shaded in blue.

How to find the area of a figure? It can be found in the “normal” way. Moreover, the area of the figure is found as the sum of the areas:

- on the segment ;

- on the segment.

That's why:

There is a more rational solution: it consists of switching to inverse functions and integrating along the axis.

How to get to inverse functions? Roughly speaking, you need to express “x” through “y”. First, let's look at the parabola:

This is enough, but let’s make sure that the same function can be derived from the lower branch:

It's easier with a straight line:

Now look at the axis: please periodically tilt your head to the right 90 degrees as you explain (this is not a joke!). The figure we need lies on the segment, which is indicated by the red dotted line. In this case, on the segment the straight line is located above the parabola, which means that the area of the figure should be found using the formula already familiar to you: . What has changed in the formula? Just a letter and nothing more.

! Note : Axis integration limits should be placedstrictly from bottom to top !

Finding the area:

On the segment, therefore:

Please note how I carried out the integration, this is the most rational way, and in the next paragraph of the task it will be clear why.

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

The original integrand function is obtained, which means the integration was performed correctly.

Answer:

2) Let us calculate the volume of the body formed by the rotation of this figure around the axis.

I’ll redraw the drawing in a slightly different design:

So, the figure shaded in blue rotates around the axis. The result is a “hovering butterfly” that rotates around its axis.

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

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

We rotate the figure circled in red around the axis, resulting in a truncated cone. Let us denote this volume by .

Rotate the circled figure green, around the axis and denoted by the volume of the resulting body of rotation.

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

We use the formula to find the volume of a body of revolution:

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

But the advantage of integration, which I recently talked about, is much easier to find , rather than first raising the integrand to the 4th power.

Answer:

Note that if the same flat figure is rotated around the axis, you will get a completely different body of rotation, with a different volume, naturally.

Example 7

Calculate the volume of a body formed by rotation around the axis of a figure bounded by curves and .

Solution: Let's make a drawing:

Along the way, we get acquainted with the graphs of some other functions. Here is an interesting graph of an even function...

For the purpose of finding the volume of a body of revolution, it is enough to use the right half of the figure, which I shaded in blue. Both functions are even, their graphs are symmetrical about the axis, and our figure is symmetrical. Thus, the shaded right part, rotating around the axis, will certainly coincide with the left unshaded part.

The volume of a body of revolution can be calculated using the formula:

In the formula, the number must be present before the integral. So it happened - everything that revolves in life is connected with this constant.

I think it’s easy to guess how to set the limits of integration “a” and “be” from the completed drawing.

Function... what is this function? Let's look at the drawing. The plane figure is bounded by the graph of the parabola at the top. This is the function that is implied in the formula.

In practical tasks, a flat figure can sometimes be located below the axis. This does not change anything - the function in the formula is squared: , thus the volume of a body of revolution is always non-negative, which is very logical.

Let's calculate the volume of a body of rotation using this formula:

As I already noted, the integral almost always turns out to be simple, the main thing is to be careful.

Answer:

In your answer, you must indicate the dimension - cubic units. That is, in our body of rotation there are approximately 3.35 “cubes”. Why cubic units? Because the most universal formulation. There could be cubic centimeters, there could be cubic meters, there could be cubic kilometers, etc., that’s how many green men your imagination can put in a flying saucer.

Example 2

Find the volume of a body formed by rotation around the axis of a figure bounded by lines , ,

This is an example for you to solve on your own. Full solution and answer at the end of the lesson.

Let's consider two more complex problems, which are also often encountered in practice.

Example 3

Calculate the volume of the body obtained by rotating around the abscissa axis of the figure bounded by the lines , , and

Solution: Let us depict in the drawing a flat figure bounded by the lines , , , , without forgetting that the equation defines the axis:

The desired figure is shaded in blue. When it rotates around its axis, it turns out to be a surreal donut with four corners.

Let us calculate the volume of the body of rotation as difference in volumes of bodies.

First, let's look at the figure circled in red. When it rotates around an axis, a truncated cone is obtained. Let us denote the volume of this truncated cone by .

Consider the figure that is circled in green. If you rotate this figure around the axis, you will also get a truncated cone, only a little smaller. Let's denote its volume by .

And, obviously, the difference in volumes is exactly the volume of our “donut”.

We use the standard formula to find the volume of a body of revolution:

1) The figure circled in red is bounded above by a straight line, therefore:

2) The figure circled in green is bounded above by a straight line, therefore:

3) Volume of the desired body of revolution:

Answer:

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

The decision itself is often written shorter, something like this:

Now let’s take a little rest and tell you about geometric illusions.

People often have illusions associated with volumes, which were noticed by Perelman (not that one) in the book Entertaining geometry. Look at the flat figure in the solved problem - it seems to be small in area, and the volume of the body of revolution is just over 50 cubic units, which seems too large. By the way, the average person drinks the equivalent of a room with an area of 18 in his entire life. square meters, which, on the contrary, seems to be too small a volume.

In general, the education system in the USSR was truly the best. The same book by Perelman, written by him back in 1950, very well develops, as the humorist said, thinking and teaches one to look for original, non-standard solutions to problems. I recently re-read some of the chapters with great interest, I recommend it, it’s accessible even for humanists. No, you don’t need to smile that I offered a free time, erudition and broad horizons in communication are a great thing.

After lyrical digression It’s just appropriate to solve a creative task:

Example 4

Calculate the volume of a body formed by rotation about the axis of a flat figure bounded by the lines , , where .

This is an example for you to solve on your own. Please note that all things happen in the band, in other words, practically ready-made limits of integration are given. Also try to draw the graphs correctly. trigonometric functions, if the argument is divided by two: , then the graphs are stretched twice along the axis. Try to find at least 3-4 points according to trigonometric tables and more accurately complete the drawing. Full solution and answer at the end of the lesson. By the way, the task can be solved rationally and not very rationally.

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

The second paragraph will be even more interesting than the first. The task of calculating the volume of a body of revolution around the ordinate axis is also a fairly common guest in test work. Along the way it will be considered problem of finding the area of a figure the second method is integration along the axis, this will allow you not only to improve your skills, but also teach you to find the most profitable solution path. There is also a practical point in this. life meaning! As my teacher on methods of teaching mathematics recalled with a smile, many graduates thanked her with the words: “Your subject helped us a lot, now we effective managers and optimally manage our staff.” Taking this opportunity, I also express my great gratitude to her, especially since I use the acquired knowledge for its intended purpose =).

Example 5

Given a flat figure bounded by the lines , , .

1) Find the area of a flat figure bounded by these lines.
2) Find the volume of the body obtained by rotating a flat figure bounded by these lines around the axis.

Attention! Even if you only want to read the second point, first Necessarily read the first one!

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

1) Let's make a drawing:

The desired figure, the area of which is to be found, is shaded in blue.

How to find the area of a figure? It can be found in the “usual” way, which was discussed in class Definite integral. How to calculate the area of a figure. Moreover, the area of the figure is found as the sum of the areas:
- on the segment ;
- on the segment.

That's why:

What's bad in this case? the usual way solutions? Firstly, we got two integrals. Secondly, integrals are roots, and roots in integrals are not a gift, and besides, you can get confused in substituting the limits of integration. In fact, the integrals, of course, are not killer, but in practice everything can be much sadder, I just selected “better” functions for the problem.

There is a more rational solution: it consists of switching to inverse functions and integrating along the axis.

How to get to inverse functions? Roughly speaking, you need to express “x” through “y”. First, let's look at the parabola:

This is enough, but let’s make sure that the same function can be derived from the lower branch:

It's easier with a straight line:

! Note: The integration limits along the axis should be set strictly from bottom to top!

Finding the area:

On the segment, therefore:

Please note how I carried out the integration, this is the most rational way, and in the next paragraph of the task it will be clear why.

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

The original integrand function is obtained, which means the integration was performed correctly.

Answer:

2) Let us calculate the volume of the body formed by the rotation of this figure around the axis.

I’ll redraw the drawing in a slightly different design:

So, the figure shaded in blue rotates around the axis. The result is a “hovering butterfly” that rotates around its axis.

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

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

We rotate the figure circled in red around the axis, resulting in a truncated cone. Let us denote this volume by .

We rotate the figure circled in green around the axis and denote it by the volume of the resulting body of rotation.

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

We use the formula to find the volume of a body of revolution:

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

But the advantage of integration, which I recently talked about, is much easier to find , rather than first raising the integrand to the 4th power.

Answer:

However, not a sickly butterfly.

Note that if the same flat figure is rotated around the axis, you will get a completely different body of rotation, with a different volume, naturally.

Example 6

Given a flat figure bounded by lines and an axis.

1) Go to inverse functions and find the area of a plane figure bounded by these lines by integrating over the variable.
2) Calculate the volume of the body obtained by rotating a flat figure bounded by these lines around the axis.

Using integrals to find the volumes of bodies of revolution

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

Specific mathematical knowledge makes it difficult to understand the principles of the device and the use of modern technology. Every person in his life has to perform quite complex calculations, use commonly used equipment, find the necessary formulas in reference books, and create simple algorithms for solving problems. IN modern society more and more specialties requiring high level education is associated with the direct application of mathematics. Thus, mathematics becomes a professionally significant subject for a student. The leading role belongs to mathematics in the formation of algorithmic thinking; it develops the ability to act according to a given algorithm and to construct new algorithms.

While studying the topic of using the integral to calculate the volumes of bodies of revolution, I suggest that students in elective classes consider the topic: “Volumes of bodies of revolution using integrals.” Below are methodological recommendations for considering this topic:

1. Area of a flat figure.

From the algebra course we know that problems of a practical nature led to the concept of a definite integral..gif" width="88" height="51">.jpg" width="526" height="262 src=">

https://pandia.ru/text/77/502/images/image006_95.gif" width="127" height="25 src=">.

To find the volume of a body of rotation formed by the rotation of a curvilinear trapezoid around the Ox axis, bounded by a broken line y=f(x), the Ox axis, straight lines x=a and x=b, we calculate using the formula

https://pandia.ru/text/77/502/images/image008_26.jpg" width="352" height="283 src=">Y

3.Cylinder volume.

https://pandia.ru/text/77/502/images/image011_58.gif" width="85" height="51">..gif" width="13" height="25">..jpg" width="401" height="355">The cone is obtained by rotating the right triangle ABC (C = 90) around the Ox axis on which leg AC lies.

Segment AB lies on the straight line y=kx+c, where https://pandia.ru/text/77/502/images/image019_33.gif" width="59" height="41 src=">.

Let a=0, b=H (H is the height of the cone), then Vhttps://pandia.ru/text/77/502/images/image021_27.gif" width="13" height="23 src=">.

5.Volume of a truncated cone.

A truncated cone can be obtained by rotating rectangular trapezoid ABCD (CDOx) around the Ox axis.

The segment AB lies on the straight line y=kx+c, where , c=r.

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

Thus, the straight line looks like https://pandia.ru/text/77/502/images/image027_17.gif" width="303" height="291 src=">

Let a=0, b=H (H is the height of the truncated cone), then https://pandia.ru/text/77/502/images/image030_16.gif" width="36" height="17 src="> = .

6. Volume of the ball.

The ball can be obtained by rotating a circle with 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="13" height="16 src=">x R.

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The volume of a body formed by rotation around an online axis. Calculation of volumes of bodies of revolution using a definite integral