The volume of a body obtained by rotation around an axis. How to calculate the volume of a body of revolution using a definite integral? Formula for the volume of a pyramid

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 revolution using this formula:

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

Answer:

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

Example 2

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

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

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

Example 3

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

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

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

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

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

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

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

We use the standard formula to find the volume of a body of 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.

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

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

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

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

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

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

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

Answer:

Example 2

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

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

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

Example 3

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

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

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

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

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

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

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

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

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

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

3) Volume of the desired body of revolution:

Answer:

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

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

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

People often have illusions associated with volumes, which were noticed by Perelman (not that one) in the book Entertaining geometry. Look at the flat figure in the solved problem - it seems to be small in area, and the volume of the body of revolution is just over 50 cubic units, which seems too large. By the way, the average person drinks the equivalent of a room 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, written by him back in 1950, very well develops, as the humorist said, thinking and teaches one to look for original, non-standard solutions to problems. I recently re-read some of the chapters with great interest, I recommend it, it’s accessible even for humanists. No, you don’t need to smile that I offered a free time, erudition and broad horizons in communication are a great thing.

After 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 the lines , , where .

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

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

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

Example 5

Given a flat figure bounded by the lines , , .

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

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

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

1) Let's make a drawing:

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

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

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

That's why:

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

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

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

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

It's easier with a straight line:

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

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

Finding the area:

On the segment, therefore:

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

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

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

Answer:

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

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

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

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

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

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

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

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

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

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

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

Answer:

However, not a sickly butterfly.

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

Example 6

Given a flat figure bounded by lines and an axis.

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

This is an example for you to solve on your own. Those interested can also find the area of a figure in the “usual” way, thereby checking point 1). But if, I repeat, you rotate a flat figure around the axis, you will get a completely different body of rotation with a different volume, by the way, the correct answer (also for those who like to solve problems).

The complete solution to the two proposed points of the task is at the end of the lesson.

Yes, and don’t forget to tilt your head to the right to understand the bodies of rotation and the limits of integration!

I was about to finish the article, but today they brought an interesting example just for finding the volume of a body of revolution around the ordinate axis. Fresh:

Example 7

Calculate the volume of a body formed by rotation around the axis of a figure bounded by curves and. The left unused branch of the parabola corresponds to the inverse function - the graph of the function is located on the segment above the axis;

It is logical to assume that the volume of a body of revolution should be sought as the sum of the volumes of bodies of revolution!

We use the formula:

In this case:

Answer:

IN problem of finding the area of a figure summation of areas is often used, but summation of volumes of bodies of rotation is apparently rare, since such a variety almost fell out of my field of vision. Still, it’s good that the example we discussed turned up in a timely manner – we managed to extract a lot of useful information.

Successful promotion of figures!

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

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 curvilinear 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 rotation

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 derivative

If the curve is given parametrically by ur-mix=x(t) ,y= t(t) f-iix’(t), y’(t), x(t), y(t) are defined on the interval [a; b], x(a)= a, x(b)= bthen making the replacement with a changex= x(t)

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

If the curve equation is specified in the polar coordinate system

Sthe surface of rotation around the axis will be equal to

Sections: Mathematics

Lesson type: combined.

Objective 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.

Lesson progress

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: “Everything is 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. “Everything is in your hands.”

II. Repetition of previously studied material.

– Let’s remember the main points of the previously studied material. To do this, let's complete the task “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 clues in the form of arrows.

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

Slide. “Map instruction” Calculation of volumes.

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

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

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

1. around the OX axis.

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

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

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

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

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

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

– Consider the following tasks

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

Answer: 1163 cm 3 .

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

IV. Consolidating new material

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

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

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

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

- Let's consider the problem.

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

cube units

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

1st group.

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

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

Error! The bookmark is not defined.

– Please tell me where we meet this figure?

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

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

All pipes are similar to a cylinder.
And I gave an example like this -
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 me more about the cone.
– The funnel is in the form of a watering can cone.
Come on, find her for me quickly.
I couldn't find the funnel
But mom made a bag,
I wrapped the cardboard around my finger
And she deftly secured it with a paper clip.
The oil is flowing, mom is happy,
The cone came out just right.

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

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

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

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

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

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

- Well, now let's rest a little.

Find a pair.

Mathematical domino melody plays.

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

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

Tests for strong students and mathematical football.

Math simulator.

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

A) an indefinite integral,

B) function,

B) differentiation.

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

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

Reflection.

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

1st line – topic name (one noun).

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

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

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

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

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

Conclusion (slide).

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

Grading. (With commentary.)

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

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

A cylinder is a simple geometric body obtained by rotating a rectangle around one of its sides. Another definition: a cylinder is a geometric body bounded by a cylindrical surface and two parallel planes that intersect it.

cylinder volume formula

If you want to know how to calculate the volume of a cylinder, then all you need to do is find the height (h) and radius (r) and plug them into the formula:

If you look closely at this formula, you will notice that (\pi r^2) is the formula for the area of a circle, and in our case, the area of the base.

Therefore, the formula for the volume of a cylinder can be written in terms of the base area and height:

Our online calculator will help you calculate the volume of a cylinder. Simply enter the specified parameters of the cylinder and get its volume.

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Volume of a cylinder formula (using base radius and height)

(V=\pi r^2 h), where

r is the radius of the cylinder base,

h - cylinder height

Volume of a cylinder formula (via base area and height)

S is the area of the cylinder base,

h - cylinder height

Cylinder volume calculator online

How to find the volume of a body of revolution using an integral

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.

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

Similarly, the volume v of a body obtained by rotation around the ordinate axis (Oy) of a curvilinear 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.

Find the volume of the body formed by rotation around the abscissa axis (Ox) of the figure bounded by the hyperbola, the abscissa axis and the 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 of radius R.

Solution. Let us consider a ball as a body obtained by rotating around the abscissa axis of a semicircle of radius R with its 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. Consequently,

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

Let us imagine the required volume as the difference in the volumes of bodies obtained by rotating the curvilinear trapezoids ABCDE and ABFDE around the abscissa axis. We find the volumes of these bodies using formula (1), in which the limits of integration are equal to and are the abscissas of points B and D of the intersection of the 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 the Ox axis (Fig.

Formulas for areas and volumes of geometric figures

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 Ox axis.

The equation of the circle LBCD is

and the equation of the BCD curve

and the equation of the BLD curve

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

Example 5.

Find the volume of the body formed by rotation around the ordinate axis (Oy) of the figure bounded by the lines and.

Let us imagine the required volume as the difference between the volumes of bodies obtained by rotating around the ordinate axis of the triangle OBA and the curvilinear trapezoid OnBA.

We find the volumes of these bodies using formula (2). The limits of integration are and - the ordinates of points O and B of the intersection of the parabola and the straight line.

Thus, we obtain the volume of the body:

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Take the test on the topic Integral

Beginning of the topic “Integral”

Indefinite integral: basic concepts, properties, table of indefinite integrals

Find the indefinite integral: beginnings, examples of solutions

Method for changing a variable in an indefinite integral

Integration by subsuming the differential sign

Method of integration by parts

Integrating Fractions

Integration of rational functions and the method of undetermined coefficients

Integration of some irrational functions

Integrating trigonometric functions

Definite integral

Area of a plane figure using an integral

Improper integrals

Calculation of double integrals

Arc length of a curve using integral

Surface area of revolution using integral

Determining the work of a force using an integral

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Volume of a geometric figure- a quantitative characteristic of the space occupied by a body or substance. The volume of a vessel's body or container is determined by its shape and linear dimensions.