The volume of a body of revolution around the x-axis is parametric. Calculation of the area of a figure bounded by a parametrically defined curve. Calculation of the volume of a body formed by the rotation of a flat figure around an axis

Sections: Mathematics

Lesson type: combined.

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

Tasks:

consolidate the ability to select curvilinear trapezoids from a number of geometric shapes 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;
contribute to the development logical thinking, competent mathematical speech, accuracy in the construction of drawings;
to cultivate interest in the subject, to operate with mathematical concepts and images, to cultivate the will, independence, perseverance in achieving the final result.

During the classes

I. Organizational moment.

Group greeting. Communication to students of the objectives of the lesson.

Reflection. Calm melody.

I would like to start today's lesson with a parable. “There was a wise man who knew everything. One person wanted to prove that the sage does not know everything. Clutching the 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, if the dead one says, I will let her out.” The sage, thinking, answered: "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 later life and in practical activities. "All in your hands".

II. Repetition of previously learned material.

Let's review the main points of the previously studied material. To do this, let's do the task "Remove the redundant word."(Slide.)

(The student goes to I.D. with the help of an eraser removes the extra word.)

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

- Let's remember the main stages and concepts related to integral calculus ..

"Mathematical bunch".

Exercise. Restore passes. (The student comes out and writes the necessary words with a pen.)

- We will hear a report on the application of integrals later.

Work in notebooks.

– The Newton-Leibniz formula was developed 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 that nature itself speaks.

– Consider how this formula is used in solving practical tasks.

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

Solution: Build on coordinate plane function graphs . Select the area of the figure 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 meet not only with flat figures, but also with three-dimensional ones, but how to calculate the volume of such bodies? For example, the volume of a planet, a comet, a meteorite, etc.

– Think about the volume and building houses, and pouring water from one vessel to another. Rules and methods for calculating volumes should have arisen, another thing is how accurate and justified they were.

Student message. (Tyurina Vera.)

The year 1612 was very fruitful for the inhabitants of the Austrian city of Linz, where the then 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 marked the beginning of a whole stream of research, which culminated in the last quarter of the 17th century. design in the works of I. Newton and G.V. Leibniz differential and integral calculus. Since that time, the mathematics of magnitude variables has taken a leading place in the system of mathematical knowledge.

- So today we will be engaged in such practical activities, therefore,

The topic of our lesson: "Calculation of the volumes of bodies of revolution using a definite integral." (Slide)

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

"Labyrinth".

Labyrinth (Greek word) means passage to the dungeon. A labyrinth is an intricate network of paths, passages, rooms that communicate with each other.

But the definition “crashed”, there were hints in the form of arrows.

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

Slide. “Instruction card” Calculation of volumes.

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

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

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

1. around the x-axis.

2. , if the rotation of the curvilinear trapezoid around the y-axis.

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

The teacher explains the solution of the examples on the blackboard.

Consider an excerpt from the famous fairy tale by A. S. Pushkin “The Tale of Tsar Saltan, of his glorious and mighty son Prince Gvidon Saltanovich and the beautiful Princess Lebed” (Slide 4):

…..
And brought a drunken messenger
On the same day, the order is:
“The tsar orders his boyars,
Wasting no time,
And the queen and the offspring
Secretly cast into the abyss of waters.”
There is nothing to do: the boyars,
Having mourned about the sovereign
And the young queen
A crowd came to her bedroom.
Declared the royal will -
She and her son have an evil fate,
Read the decree aloud
And the queen at the same time
They put me in a barrel with my son,
Prayed, rolled
And they let me into the okian -
So ordered de Tsar Saltan.

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 y-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 y = , x = 4, y = 0.

IV. Fixing new material

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

Let's plot the graphs of the function. y=x2, y2=x. Schedule y 2 = x transform to the form y= .

We have V \u003d V 1 - V 2 Let's calculate the volume of each function

- Now, let's look at the tower for a radio station in Moscow on Shabolovka, built according to the project of a wonderful Russian engineer, honorary academician V. G. Shukhov. It consists of parts - hyperboloids of revolution. Moreover, each of them is made of rectilinear metal rods connecting adjacent circles (Fig. 8, 9).

- Consider the problem.

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

cube units

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

1st group.

Hit! Hit! Another hit!
A ball flies into the gate - BALL!
And this is a watermelon ball
Green, round, delicious.
Look better - what a ball!
It is made up of circles.
Cut into circles watermelon
And taste them.

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

Error! The bookmark is not defined.

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

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

"Cylinder - what is it?" I asked my dad.
The father laughed: The top hat is a hat.
To have representation is correct,
The cylinder, let's say, is a tin can.
The pipe of the steamer is a 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.
It also looks like a cylinder.

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

2nd group. CONE (slide).

Mom said: And now
About the cone will be my story.
Stargazer in a high cap
Counts the stars all year round.
CONE - stargazer's hat.
That's what he is. Understood? That's it.
Mom was at the table
She poured oil into bottles.
- Where is the funnel? No funnel.
Look. Don't stand on the sidelines.
- Mom, I will not move from the place,
Tell me more about the cone.
- The funnel is in the form of a cone of a watering can.
Come on, find me quickly.
I couldn't find the funnel
But mom made a bag,
Wrap cardboard around your finger
And deftly fastened with a paper clip.
Oil is pouring, mom is happy
The cone came out just right.

Exercise. Calculate the volume of the body obtained by rotation around the x-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 mystery and mystery in it.
The Spasskaya Tower on Red Square
Both children and adults are well known.
Look at the tower - ordinary in appearance,
What's on top of her? Pyramid!

Exercise. Homework plot a 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 the integral.

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

"Now let's get some rest."

Find a couple.

Mathematical domino melody plays.

“The road that he himself was looking for will never be forgotten ...”

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

Tests for strong learners and math 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 revolution.

Reflection.

Acceptance of reflection in the form cinquain(five lines).

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

2nd line - a description of the topic in a nutshell, two adjectives.

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

4th line - a phrase of four words, 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, rotate, calculate.
A body obtained by rotating a curvilinear trapezoid (around its base).
Body of revolution (3D geometric body).

Conclusion (slide).

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

Grading. (With commentary.)

The great Omar Khayyam is a mathematician, poet, and philosopher. He calls to be masters of his destiny. Listen to an excerpt from his work:

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

Let us find the volume of the body generated by the rotation of the cycloid arch around its base. Roberval found it by breaking the resulting egg-shaped body (Fig. 5.1) into infinitely thin layers, inscribing cylinders into these layers and adding up their volumes. The proof is long, tedious, and not entirely rigorous. Therefore, to calculate it, we turn to higher mathematics. Let us set the cycloid equation parametrically.

In integral calculus, when studying volumes, he uses the following remark:

If the curve bounding the curvilinear trapezoid is given by parametric equations and the functions in these equations satisfy the conditions of the theorem on the change of variable in a certain integral, then the volume of the body of rotation of the trapezoid around the Ox axis will be calculated by the formula:

Let's use this formula to find the volume we need.

In the same way, we calculate the surface of this body.

L=((x,y): x=a(t - sin t), y=a(1 - cost), 0 ? t ? 2р)

In integral calculus, there is the following formula for finding the surface area of a body of revolution around the x-axis of a curve specified on a segment parametrically (t 0 ?t ?t 1):

Applying this formula to our cycloid equation, we get:

Consider also another surface generated by the rotation of the cycloid arc. To do this, we will build a mirror reflection of the cycloid arch relative to its base, and we will rotate the oval figure formed by the cycloid and its reflection around the KT axis (Fig. 5.2)

First, let's find the volume of the body formed by the rotation of the cycloid arch around the KT axis. Its volume will be calculated by the formula (*):

Thus, we calculated the volume of half of this turnip body. Then the total volume will be

Consider examples of applying the obtained formula, which allows you to calculate the areas of figures bounded by parametrically specified lines.

Example.

Calculate the area of a figure bounded by a line whose parametric equations look like .

Solution.

In our example, the parametrically defined line is an ellipse with semi-axes of 2 and 3 units. Let's build it.

Find the area of a quarter of the ellipse located in the first quadrant. This area lies in the interval . We calculate the area of the entire figure by multiplying the resulting value by four.

What we have:

For k = 0 we get the interval . On this interval, the function monotonically decreasing (see section ). We apply the formula to calculate the area and find the definite integral using the Newton-Leibniz formula:

So the area of the original figure is .

Comment.

A logical question arises: why did we take a quarter of the ellipse, and not half? It was possible to consider the upper (or lower) half of the figure. She is in the range . For this case, we would have

That is, for k = 0 we get the interval . On this interval, the function monotonically decreasing.

Then the area of half of the ellipse is given by

But the right or left halves of the ellipse cannot be taken.

The parametric representation of an ellipse centered at the origin and semi-axes a and b has the form . If we act in the same way as in the parsed example, we get formula for calculating the area of an ellipse .

A circle centered at the origin of coordinates of radius R is given by a system of equations through the parameter t. If we use the obtained formula for the area of an ellipse, then we can immediately write formula for finding the area of a circle radius R : .

Let's solve one more example.

Example.

Calculate the area of a figure bounded by a curve given parametrically.

Solution.

Looking ahead a little, the curve is an "elongated" astroid. (The astroid has the following parametric representation).

Let us dwell in detail on the construction of a curve bounding a figure. We will build it point by point. Usually such a construction is sufficient for solving most problems. In more difficult cases, no doubt, a detailed study of the parametric given function using differential calculus.

In our example .

These functions are defined for all real values of the parameter t, and, from the properties of the sine and cosine, we know that they are periodic with a period of two pi. Thus, calculating the values of functions for some (For example ), we get a set of points .

For convenience, we will enter the values in the table:

We mark the points on the plane and SEQUENTIALLY connect them with a line.

Let's calculate the area of the area located in the first coordinate quarter. For this area .

At k=0 we get the interval , on which the function decreases monotonically. We use the formula to find the area:

Received definite integrals we calculate using the Newton-Leibniz formula, and we find the antiderivatives for the Newton-Leibniz formula using a recursive formula of the form , Where .

Therefore, the area of a quarter of the figure is , then the area of the whole figure is equal to .

Similarly, one can show that astroid area located as , and the area of the figure bounded by the line is calculated by the formula .

Greetings, dear students of Argemony University!

A little more - and the course will be completed, and now we'll do this.

Zhouli slightly waved her hand - and a figure appeared in the air. Or rather, it was a rectangular trapezoid. It simply hung in the air, created by the magical energy that flowed along its sides, and also swirled inside the trapezoid itself, which made it sparkle and shimmer.
Then the teacher slightly noticeably made a circular motion with her fingers - and the trapezoid began to rotate around an invisible axis. Slowly at first, then faster and faster - so that a volumetric figure began to clearly appear in the air. It felt like magical energy was flowing through her.

Then the following happened: the sparkling contours of the figure and its interior began to fill with some substance, the glow became less and less noticeable, but the figure itself looked more and more like something tangible. Grains of material were evenly distributed over the figure. And now it's all over: both rotation and glow. An object resembling a funnel hung in the air. Zhouli gently moved it to the table.

Here you go. Something like this can materialize many objects - by rotating some flat figures around imaginary lines. Of course, for materialization, a certain amount of substance is needed, which will fill the entire volume formed and temporarily held with the help of magical energy. But in order to accurately calculate how much substance is needed, you also need to know the volume of the resulting body. Otherwise, if the substance is small, then it will not fill the entire volume and the body may turn out to be fragile, with flaws. And to materialize and still retain a large excess of matter is an unnecessary expenditure of magical energy.
But what if we have a limited amount of the substance? Then, knowing how to calculate the volumes of bodies, we can estimate what size body we can make without much expenditure of magical energy.
As for the surplus of attracted material, there is another thought. Where does the excess matter go? Are they crumbling when not used? Or stick to the body anyhow?
In general, there is still something to think about. If you have any thoughts, I would love to hear them. In the meantime, let's move on to calculating the volumes of bodies obtained in this way.
Several cases are considered here.

Case 1

The area we will be rotating is the most classic curvilinear trapezoid.

Naturally, we can only rotate it around the OX axis. If this trapezoid is moved to the right horizontally so that it does not cross the OY axis, then it can be rotated about this axis. The incantation formulas for both cases are as follows:

You and I have already mastered the basic magical effects on functions quite well, so I think it will not be difficult for you, if necessary, to move the figure in such a way in the coordinate axes that it is located conveniently for working with it.

Case 2

You can rotate not only the classic curvilinear trapezoid, but also a figure like this:

When rotating, we get a kind of ring. And by moving the figure to the positive area, we can also rotate it about the OY axis. We will also get a ring or not. It all depends on how the figure will be located: if its left border passes exactly along the OY axis, then the ring will not work. You can calculate the volumes of such bodies of revolution using the following spells:

Case 3

Recall that we have wonderful curves, but they are not set in the usual way, but in a parametric form. Such curves are often closed. The parameter t must be changed in such a way that the closed figure remains on the left when traversing it along the curve (boundary).

Then, to calculate the volumes of bodies of rotation relative to the OX or OY axis, you need to use the following spells:

The same formulas can also be used for the case of non-closed curves: when both ends lie on the OX axis or on the OY axis. The figure somehow turns out to be closed: the ends are closed by a segment of the axis.

Case 4

Some of the wonderful curves we have are given by polar coordinates (r=r(fi)). And then the figure can be rotated about the polar axis. In this case, the Cartesian coordinate system is combined with the polar one and it is assumed
x=r(fi)*cos(fi)
y=r(fi)*sin(fi)
Thus, we come to the parametric form of the curve, where the parameter fi must change so that when the curve is traversed, the area remains on the left.
And we use incantation formulas from case 3.

However, for the case of polar coordinates, there is also a spell formula:

Of course, plane figures can also be rotated about any other lines, not only about the OX and OY axes, but these manipulations are already more complex, so we will restrict ourselves to the cases that were considered in the lecture.

And now homework . I will not give you specific figures. We have already learned many functions, and I would like you to construct something yourself that you may need in magical practice. I think four examples for all the cases mentioned in the lecture will be enough.

Before proceeding to the formulas for the area of a surface of revolution, we give a brief formulation of the surface of revolution itself. The surface of revolution, or, what is the same, the surface of a body of revolution is a spatial figure formed by the rotation of a segment AB curve around the axis Ox(picture below).

Let us imagine a curvilinear trapezoid bounded from above by the mentioned segment of the curve. The body formed by the rotation of this trapezoid around the same axis Ox, and there is a body of revolution. And the surface area of rotation or the surface of a body of rotation is its outer shell, not counting the circles formed by rotation around the axis of lines x = a And x = b .

Note that the body of revolution and, accordingly, its surface can also be formed by rotating the figure not around the axis Ox, and around the axis Oy.

Calculating the area of a surface of revolution given in rectangular coordinates

Let in rectangular coordinates on the plane by the equation y = f(x) a curve is given, the rotation of which around the coordinate axis forms a body of revolution.

The formula for calculating the surface area of revolution is as follows:

(1).

Example 1 Find the surface area of a paraboloid formed by rotation about an axis Ox the arc of the parabola corresponding to the change x from x= 0 to x = a .

Solution. We explicitly express the function that defines the arc of the parabola:

Let's find the derivative of this function:

Before using the formula for finding the area of the surface of revolution, let's write the part of its integrand that is the root and substitute the derivative we just found there:

Answer: The arc length of the curve is

Example 2 Find the area of the surface formed by rotation about an axis Ox astroids.

Solution. It is enough to calculate the surface area resulting from the rotation of one branch of the astroid, located in the first quarter, and multiply it by 2. From the astroid equation, we explicitly express the function that we will need to substitute in the formula to find the surface area of rotation:

We perform integration from 0 to a:

Calculation of the surface area of revolution given parametrically

Consider the case when the curve forming the surface of revolution is given by the parametric equations

Then the area of the surface of revolution is calculated by the formula

(2).

Example 3 Find the area of the surface of revolution formed by the rotation about an axis Oy figure bounded by a cycloid and a straight line y = a. The cycloid is given by the parametric equations