Calculation of the volume of a figure bounded by lines online. Calculation of volumes of bodies of revolution using a definite integral

Topic: "Calculation of the volumes of bodies of revolution using a 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 select curvilinear trapezoids from a row geometric shapes and work out 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.

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’ll kill her, if the dead one says, I’ll let her out.” 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 later life and in practical activities. “Everything is in your hands.”

II. Repetition of previously learned material.

Let's recall the main points of the previously studied material. To do this, we will complete the task “Delete the extra word”.

(Students say an extra word.)

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

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

The task. Restore passes. (Student comes out and writes with a marker necessary words.)

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 spoken by nature itself.

Consider how, when solving practical tasks this formula is used.

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

Solution: Let us construct on the coordinate plane the graphs of the functions . 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? (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 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, comet, meteorite, etc.

They think about the volume when 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.

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.

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 deal with such practical activities, Consequently,

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

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

"Labyrinth".

The task. 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 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.

Students write down the basic formulas in a notebook.

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

1. Find the volume of the body obtained by rotating around the y-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 abscissa axis y = , x = 4, y = 0.

Solution.

V. Math simulator.

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

BUT) 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. Fixing new material

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

Let's plot the graphs of the function. y=x2, y2=x. The graph y2 = x is transformed into the form y = .

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

Output:

The 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!

VI. Grading.(With commentary.)

Great Omar Khayyam - mathematician, poet, 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.

Using Integrals to Find Volumes of Solids 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 use modern technology. Every person in his life has to do enough complex calculations, use commonly used equipment, find in reference books to apply the necessary formulas, compose simple algorithms for solving problems. IN modern society more specialties requiring high level education is associated with the direct application of mathematics. Thus, for a schoolchild, mathematics becomes a professionally significant subject. The leading role belongs to mathematics in the formation of algorithmic thinking, it brings up the ability to act according to a given algorithm and design new algorithms.

Studying the topic of using the integral to calculate the volumes of bodies of revolution, I suggest that students in optional classes consider the topic: "Volumes of bodies of revolution using integrals." Here are some guidelines for dealing with this topic:

1. The area of a flat figure.

From the course of algebra, we know that practical problems 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 revolution 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 by the formula

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

3. The volume of the cylinder.

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 right triangle ABC(C=90) around the Ox axis on which the leg AC lies.

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

Since the line passes through the 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. The volume of the ball.

The ball can be obtained by rotating a circle with center (0;0) around the x-axis. The semicircle located above the x-axis is given by the equation

https://pandia.ru/text/77/502/images/image034_13.gif" width="13" height="16 src=">x R.

except finding the area of a flat figure using a definite integral (see 7.2.3.) the most important application of the theme is calculation of the volume of a body of revolution. The material is simple, but the reader must be prepared: it is necessary to be able to solve indefinite integrals medium complexity and apply the Newton-Leibniz formula in definite integral, n Strong drafting skills are also required. 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 revolution, the length of an arc, the surface area of \u200b\u200bthe body, and much more. Imagine some flat figure on the coordinate plane. Represented? ... Now this figure can also be rotated, and rotated in two ways:

- around the x-axis ;

- around the y-axis .

Let's take a look at both cases. The second method of rotation is especially interesting, it causes the greatest 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 the rotation of a flat figure around an axis OX

Example 1

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

Solution: As in the problem of finding the area, the solution starts with a drawing of a flat figure. That is, on the plane XOY it is necessary to construct a figure bounded by lines , while not forgetting that the equation defines the axis . The drawing here is pretty simple:

The desired flat figure is shaded in blue, it is she who rotates around the axis. As a result of rotation, such a slightly egg-shaped flying saucer with two sharp peaks on the axis is obtained. 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 the body is formed as a result of rotation around an axisOX, it is mentally divided into parallel layers of small thickness dx that are perpendicular to the axis OX. The volume of the whole 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 high dx and with base radius f(x). Then the volume of one layer is the product of the base area π f 2 to the height of the cylinder ( dx), or π∙ f 2 (x)∙dx. And the area of the entire body of revolution is the sum of elementary volumes, or the corresponding definite integral. The volume of a body of revolution can be calculated by the formula:

How to set the integration limits "a" and "be" is easy to guess from the completed drawing. Function... what is this function? Let's look at the drawing. The flat figure is bounded by the parabola graph from above. 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 quite logical. Calculate the volume of the body of revolution 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 the answer, it is necessary to indicate the dimension - cubic units. That is, in our body of rotation there are approximately 3.35 "cubes". Why exactly cubic units? Because it is the most universal formulation. There may be cubic centimeters, there may be cubic meters, there may be cubic kilometers, etc., that's how many little green men your imagination can fit into a flying saucer.

Example 2

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

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

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 lines , , , , while not forgetting that the equation x= 0 specifies the axis OY:

The desired figure is shaded in blue. When it rotates around the axis OX it turns out a flat angular bagel (a washer with two conical surfaces).

The volume of the body of revolution is calculated as body volume difference. First, let's look at the figure that is circled in red. When it rotates around the axis OX resulting in a truncated cone. Let us denote the volume of this truncated cone as V 1 .

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

Obviously, the volume difference V = V 1 - V 2 is the volume of our "donut".

We use the standard formula for finding the volume of a body of revolution:

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

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

3) The 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 made shorter, something like this:

Definition 3. A body of revolution is a body obtained by rotating a flat figure around an axis that does not intersect the figure and lies in the same plane with it.

The axis of rotation can also intersect the figure if it is the axis of symmetry of the figure.

Theorem 2.
, axis
and straight line segments
And

rotates around an axis
. Then the volume of the resulting body of revolution can be calculated by the formula

(2)

Proof. For such a body, the section with the abscissa is a circle of radius
, means
and formula (1) gives the desired result.

If the figure is limited by the graphs of two continuous functions
And
, and line segments
And
, moreover
And
, then when rotating around the abscissa axis, we get a body whose volume

Example 3 Calculate the volume of a torus obtained by rotating a circle bounded by a circle

around the x-axis.

R solution. The specified circle is bounded from below by the graph of the function
, and above -
. The difference of the squares of these functions:

Desired volume

(the graph of the integrand is the upper semicircle, so the integral written above is the area of the semicircle).

Example 4 Parabolic segment with base
, and height , revolves around the base. Calculate the volume of the resulting body ("lemon" by Cavalieri).

R solution. Place the parabola as shown in the figure. Then its equation
, and
. Let's find the value of the parameter :
. So, the desired volume:

Theorem 3. Let a curvilinear trapezoid bounded by the graph of a continuous non-negative function
, axis
and straight line segments
And
, moreover
, rotates around an axis
. Then the volume of the resulting body of revolution can be found by the formula

(3)

proof idea. Splitting the segment
dots

, into parts and draw straight lines
. The whole trapezoid will decompose into strips, which can be considered approximately rectangles with a base
and height
.

The cylinder resulting from the rotation of such a rectangle is cut along the generatrix and unfolded. We get an “almost” parallelepiped with dimensions:
,
And
. Its volume
. So, for the volume of a body of revolution we will have an approximate equality

To obtain exact equality, we must pass to the limit at
. The sum written above is the integral sum for the function
, therefore, in the limit we obtain the integral from formula (3). The theorem has been proven.

Remark 1. In Theorems 2 and 3, the condition
can be omitted: formula (2) is generally insensitive to the sign
, and in formula (3) it suffices
replaced by
.

Example 5 Parabolic segment (base
, height ) revolves around the height. Find the volume of the resulting body.

Solution. Arrange the parabola as shown in the figure. And although the axis of rotation crosses the figure, it - the axis - is the axis of symmetry. Therefore, only the right half of the segment should be considered. Parabola equation
, and
, means
. We have for volume:

Remark 2. If the curvilinear boundary of a curvilinear trapezoid is given by the parametric equations
,
,
And
,
then formulas (2) and (3) can be used with the replacement on the
And
on the
when it changes t from
before .

Example 6 The figure is bounded by the first arc of the cycloid
,
,
, and the abscissa axis. Find the volume of the body obtained by rotating this figure around: 1) the axis
; 2) axles
.

Solution. 1) General formula
In our case:

2) General formula
For our figure:

We encourage students to do all the calculations themselves.

Remark 3. Let a curvilinear sector bounded by a continuous line
and rays
,

, rotates around the polar axis. The volume of the resulting body can be calculated by the formula.

Example 7 Part of a figure bounded by a cardioid
, lying outside the circle
, rotates around the polar axis. Find the volume of the resulting body.

Solution. Both lines, and hence the figure they limit, are symmetrical about the polar axis. Therefore, it is necessary to consider only the part for which
. The curves intersect at
And

at
. Further, the figure can be considered as the difference of two sectors, and hence the volume can be calculated as the difference of two integrals. We have:

Tasks for an independent solution.

1. A circular segment whose base
, height , revolves around the base. Find the volume of the body of revolution.

2. Find the volume of a paraboloid of revolution whose base , and the height is .

3. Figure bounded by an astroid
,
rotates around the x-axis. Find the volume of the body, which is obtained in this case.

4. Figure bounded by lines
And
rotates around the x-axis. Find the volume of the body of revolution.

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;
to promote the development of 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".

The task. 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: Let's build graphs of functions on the coordinate plane . 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.)

- 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.

The task. 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 famous fairy tale 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

Mistake! 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 a correct idea,
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 -
Kaleidoscope My love,
You can't take your eyes off him.
It also looks like a cylinder.

- The task. Homework to plot a 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.

The task. 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!

The task. 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. 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).

Output (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.
The development of modern science is 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.