The volume of a body obtained by rotation around the x axis. Using integrals to find the volumes of bodies of revolution. How to calculate the volume of a body of revolution

Using a definite integral, you can calculate not only areas of plane figures, but also the volumes of bodies formed by the rotation of these figures around coordinate axes.

Examples of such bodies are in the figure below.

In the problems we have curved trapezoids that rotate around an axis Ox or around an axis Oy. To calculate the volume of a body formed by rotation of a curved trapezoid, we need:

• number "pi" (3.14...);
• definite integral of the square of the "ig" - a function that specifies a rotating curve (this is if the curve rotates around the axis Ox );
• definite integral of the square "x", expressed from the "y" (this is if the curve rotates around the axis Oy );
• limits of integration - a And b.

So, a body that is formed by rotation around an axis Ox curvilinear trapezoid bounded above by the graph of the function y = f(x) , has volume

Same volume v body obtained by rotation around the ordinate axis ( Oy) of a curved trapezoid is expressed by the formula

When calculating the area of ​​a plane figure, we learned that the areas of some figures can be found as the difference of two integrals in which the integrands are those functions that limit the figure from above and below. This is similar to the situation with some bodies of rotation, the volumes of which are calculated as the difference between the volumes of two bodies; such cases are discussed in examples 3, 4 and 5.

Example 1.Ox) a figure bounded by a hyperbola, x-axis and straight lines,.

Solution. We find the volume of a body of rotation using formula (1), in which , and the limits of integration a = 1 , b = 4 :

Example 2. Find the volume of a sphere with radius R.

Solution. Let us consider the ball as a body obtained by rotating around the x-axis of a semicircle of radius R with center at the origin. Then in formula (1) the integrand function will be written in the form , and the limits of integration are - R And R. Hence,

Example 3. Find the volume of the body formed by rotation around the abscissa axis ( Ox) figure enclosed between parabolas and .

Solution. Let us imagine the required volume as the difference in the volumes of bodies obtained by rotating curvilinear trapezoids around the abscissa axis ABCDE And ABFDE. We find the volumes of these bodies using formula (1), in which the limits of integration are equal to and - the abscissa of the points B And D intersections of parabolas. Now we can find the volume of the body:

Example 4. Calculate the volume of a torus (a torus is a body obtained by rotating a circle of radius a around an axis lying in its plane at a distance b from the center of the circle (). For example, a steering wheel has the shape of a torus).

Solution. Let the circle rotate around an axis Ox(Fig. 20). The volume of a torus can be represented as the difference in the volumes of bodies obtained from the rotation of curvilinear trapezoids ABCDE And ABLDE around the axis Ox.

Equation of a circle LBCD looks like

and the equation of the curve BCD

and the equation of the curve BLD

Using the difference between the volumes of the bodies, we obtain for the volume of the torus v expression

Let the line be limited. a plane figure is defined in a polar coordinate system.

Example: Calculate the circumference: x 2 +y 2 =R 2

Calculate the length of the 4th part of the circle located in the first quadrant (x≥0, y≥0):

If the equation of the curve is specified in parameter form:
, the functions x(t), y(t) are defined and continuous along with their derivatives on the interval [α,β]. Derivative, then substituting into the formula:
and given that

we get
add a multiplier
under the sign of the root and we finally get

Note: Given a plane curve, you can also consider a function given a parameter in space, then add the function z=z(t) and the formula

Example: Calculate the length of the astroid, which is given by the equation: x=a*cos 3 (t), y=a*sin 3 (t), a>0

Calculate the length of the 4th part:

according to the formula

Arc length of a plane curve specified in a polar coordinate system:

Let the curve equation be given in the polar coordinate system:
- a continuous function, together with its derivative on the interval [α,β].

Formulas for transition from polar coordinates:

consider as parametric:

ϕ - parameter, according to f-le

2

Ex: Calculate the length of the curve:
>0

Concept: let's calculate half the circumference:

The volume of a body, calculated from the cross-sectional area of ​​the body.

Let a body be given, bounded by a closed surface, and let the area of ​​any section of this body be known by a plane perpendicular to the Ox axis. This area will depend on the position of the cutting plane.

let the whole body be enclosed between 2 planes perpendicular to the Ox axis, intersecting it at points x=a, x=b (a

To determine the volume of such a body, we divide it into layers using cutting planes perpendicular to the Ox axis and intersecting it at points. In every partial interval
. Let's choose

and for each value i=1,….,n we will construct a cylindrical body, the generatrix of which is parallel to Ox, and the guide is the contour of the section of the body by plane x=C i, the volume of such an elementary cylinder with base area S=C i and height ∆x i . V i =S(C i)∆x i . The volume of all such elementary cylinders will be
. The limit of this sum, if it exists and is finite at max ∆х  0, is called the volume of the given body.

.

Since V n is the integral sum for a function S(x) continuous on an interval, then the indicated limit exists (the conditions of existence) and is expressed by def. Integral.

- volume of the body, calculated from the cross-sectional area.

Volume of the body of rotation:

Let the body be formed by rotation around the Ox axis of a curvilinear trapezoid limited by the graph of the function y=f(x), the Ox axis and the straight lines x=a, x=b.
Let the function y=f(x) be defined and continuous on the segment and non-negative on it, then the section of this body by a plane perpendicular to Ox is a circle with radius R=y(x)=f(x). Area of ​​the circle S(x)=Пy 2 (x)=П 2. Substituting the formula

we obtain a formula for calculating the volume of a body of rotation around the Ox axis:

If a curvilinear trapezoid, limited by the graph of a continuous function, rotates around the Oy axis, then the volume of such a body of rotation is:
The same volume can be calculated using the formula:

. If the line is given by parametric equations:

By replacing the variable we get:

If the line is given by parametric equations:

y (α)= c , y (β)= d . Making the replacement y = y (t) we get: .

Calculate the bodies of revolution around the axis of the parabola, 2) Calculate V of a body of revolution around the OX axis of a curved trapezoid bounded by a straight line y=0, an arc

(with center at point(1;0), and radius=1), with .

Surface area of ​​a body of revolution

Let a given surface be formed by rotating the curve y =f(x) around the Ox axis. It is necessary to determine S of this surface at .

Let the function y =f(x) be defined and continuous, have an unnatural and non-negative at all points of the segment [a;b]

Let us draw chords of length which we denote respectively (n-chords)

according to Lagrange's theorem:

The surface area of ​​the entire described broken line will be equal to

Definition: the limit of this sum, if it is finite, when the largest link of the broken line max, is called the area of ​​the surface of revolution under consideration.

It can be proven that one hundred the limit of the sum is equal to the limit of the integrated sum for p-th

Formula for S surface of a body of revolution =

S of the surface formed by Rotation of the arc of the curve x=g(x) around the Oy axis at

Continuous with its derivativexIf the curve is given parametrically by ur-miy= =x(t) ,(=x(t) ,) tx’(=x(t) ,), y’(=x(t) ,), x(=x(t) ,), y(=x(t) ,f-iia; b], x(a)= a, x(b)= b) are defined on the interval [x= x(=x(t) ,)

then making the replacement with a change

If the curve is given parametrically, making a change in the formula we get:

If the curve equation is specified in the polar coordinate systemS

I. Volumes of bodies of revolution. 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

Using integrals to find the volumes of bodies of rotation

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 that require a high level of education are 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. One of them is calculating the area of ​​a flat figure bounded by a continuous line y=f(x) (where f(x)DIV_ADBLOCK243">

Let's calculate the area of ​​a curvilinear trapezoid using the formula if the base of the trapezoid lies on the x-axis or using the formula https://pandia.ru/text/77/502/images/image004_49.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 a 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.

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 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 palms, 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 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 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 revolution 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 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 -
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 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.