Calculation of areas of volumes of bodies of revolution. How to calculate the volume of a body of revolution? Work in notebooks

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 flat figure is bounded by the parabola graph 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 integrand in the formula is squared: thus the integral 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 revolution 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.

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 rotation:

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 rotation:

Answer:

It is interesting 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 was noticed by Perelman (another) 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 of 18 square meters of liquid in his entire life, which, on the contrary, seems too small a volume.

In general, the education system in the USSR was truly the best. The same book by Perelman, published back in 1950, very well develops, as the humorist said, thinking and teaches you 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 a lyrical digression, it is 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 lines,, where.

This is an example for you to solve on your own. Please note that all cases occur in the band, in other words, ready-made limits of integration are actually given. Draw the graphs of trigonometric functions correctly, let me remind you of the lesson material about geometric transformations of graphs : if the argument is divided by two: , then the graphs are stretched along the axis twice. It is advisable to find at least 3-4 points according to trigonometric tables to complete the drawing more accurately. Full solution and answer at the end of the lesson. By the way, the task can be solved rationally and not very rationally.

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 rotation;
• 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.)

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. (Slide 2)

– 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 hints 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 a 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: the 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 -
My beloved kaleidoscope,
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.

1. Volume.
2. Definite integral, integrable function.
3. We build, we rotate, we calculate.
4. A body obtained by rotating a curved trapezoid (around its base).
5. 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.

Except finding the area of ​​a plane figure using a definite integral (see 7.2.3.) the most important application of the topic is calculating the volume of a body of rotation. The material is simple, but the reader must be prepared: you must be able to solve indefinite integrals medium complexity and apply the Newton-Leibniz formula in definite integral, n You also need strong drawing skills. In general, there are many interesting applications in integral calculus; using a definite integral, you can calculate the area of ​​a figure, the volume of a body of rotation, the length of an arc, the surface area of ​​a body and much more. Imagine some flat figure on the coordinate plane. Introduced? ... Now this figure can also be rotated, and rotated in two ways:

– around the x-axis ;

– around the ordinate axis .

Let's look at both cases. The second method of rotation is especially interesting; it causes the most difficulties, but in fact the solution is almost the same as in the more common rotation around the x-axis. Let's start with the most popular type of rotation.

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

Example 1

Calculate the volume of a body obtained by rotating a figure bounded by lines around an axis.

Solution: As in the problem of finding the area, the solution begins with a drawing of a flat figure. That is, on a plane XOY it is necessary to construct a figure bounded by the lines , and do not forget that the equation specifies the axis. The drawing here is quite simple:

The desired flat figure is shaded in blue; it is the one that rotates around the axis. As a result of rotation, the result is a slightly ovoid flying saucer with two sharp peaks on the axis OX, symmetrical about the axis OX. In fact, the body has a mathematical name, look in the reference book.

How to calculate the volume of a body of revolution? If a body is formed as a result of rotation around an axisOX, it is mentally divided into parallel layers of small thickness dx, which are perpendicular to the axis OX. The volume of the entire body is obviously equal to the sum of the volumes of such elementary layers. Each layer, like a round slice of lemon, is a low cylinder in height dx and with base radius f(x). Then the volume of one layer is the product of the base area π f 2 per cylinder height ( dx), or π∙ f 2 (x)∙dx. And the area of ​​the entire body of rotation is the sum of elementary volumes, or the corresponding definite integral. The volume of a body of revolution can be calculated using the formula:

.

How to set the limits of integration “a” and “be” can be easily guessed 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 OX. This does not change anything - the function in the formula is squared: f 2 (x), 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 we have 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 this is 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 an axis OX 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.

Example 3

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

Solution: Let us depict in the drawing a flat figure bounded by the lines , , , , without forgetting that the equation x= 0 specifies the axis OY:

The desired figure is shaded in blue. When it rotates around an axis OX the result is a flat, angular donut (a washer with two conical surfaces).

Let us calculate the volume of the body of revolution as difference in volumes of bodies. First, let's look at the figure circled in red. When it rotates around an axis OX the result is a truncated cone. Let us denote the volume of this truncated cone by V 1 .

Consider the figure that is circled in green. If you rotate this figure around the axis OX, then you get the same truncated cone, only a little smaller. Let us denote its volume by V 2 .

It is obvious that the difference in volumes V = V 1 - V 2 is 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 interesting 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:

I. Volumes of bodies of rotation. Preliminarily study Chapter XII, paragraphs 197, 198 from the textbook by G. M. Fikhtengolts * Analyze in detail the examples given in paragraph 198.

508. Calculate the volume of a body formed by rotating an ellipse around the Ox axis.

Thus,

530. Find the surface area formed by rotation around the Ox axis of the sinusoid arc y = sin x from point X = 0 to point X = It.

531. Calculate the surface area of ​​a cone with height h and radius r.

532. Calculate the surface area formed

rotation of the astroid x3 -)- y* - a3 around the Ox axis.

533. Calculate the surface area formed by rotating the loop of the curve 18 ug - x (6 - x) z around the Ox axis.

534. Find the surface of the torus produced by the rotation of the circle X2 - j - (y-3)2 = 4 around the Ox axis.

535. Calculate the surface area formed by the rotation of the circle X = a cost, y = asint around the Ox axis.

536. Calculate the surface area formed by the rotation of the loop of the curve x = 9t2, y = St - 9t3 around the Ox axis.

537. Find the surface area formed by rotating the arc of the curve x = e*sint, y = el cost around the Ox axis

from t = 0 to t = —.

538. Show that the surface produced by the rotation of the cycloid arc x = a (q> -sin φ), y = a (I - cos φ) around the Oy axis is equal to 16 u2 o2.

539. Find the surface obtained by rotating the cardioid around the polar axis.

540. Find the surface area formed by the rotation of the lemniscate Around the polar axis.

Additional tasks for Chapter IV

Areas of plane figures

541. Find the entire area of ​​the region bounded by the curve And the axis Ox.

542. Find the area of ​​the region bounded by the curve

And the axis Ox.

543. Find the part of the area of ​​the region located in the first quadrant and bounded by the curve

l coordinate axes.

544. Find the area of ​​the region contained inside

loops:

545. Find the area of ​​the region bounded by one loop of the curve:

546. Find the area of ​​the region contained inside the loop:

547. Find the area of ​​the region bounded by the curve

And the axis Ox.

548. Find the area of ​​the region bounded by the curve

And the axis Ox.

549. Find the area of ​​the region bounded by the Oxr axis

straight and curve