Calculate the volume of a body of revolution around an axis online. How to calculate the volume of a body of revolution using a definite integral

Let T be a body of revolution formed by rotation around the x-axis curvilinear trapezoid, located in the upper half-plane and bounded by the x-axis, straight lines x=a and x=b and graph continuous function y=f(x) .

Let's prove that this the body of revolution is cubable and its volume is expressed by the formula

V=\pi \int\limits_(a)^(b) f^2(x)\,dx= \pi \int\limits_(a)^(b)y^2\,dx\,.

First, we prove that this body of revolution is regular if we take as \Pi the plane Oyz perpendicular to the axis of revolution. Note that the section located at a distance x from the plane Oyz is a circle of radius f(x) and its area S(x) is \pi f^2(x) (Fig. 46). Therefore, the function S(x) is continuous due to the continuity of f(x) . Next, if S(x_1)\leqslant S(x_2), then this means that . But the projections of the sections onto the plane Oyz are circles of radii f(x_1) and f(x_2) with center O , and from f(x_1)\leqslant f(x_2) it follows that the circle of radius f(x_1) is contained in the circle of radius f(x_2) .

So, the body of rotation is regular. Therefore, it is cubeable and its volume is calculated by the formula

V=\pi \int\limits_(a)^(b) S(x)\,dx= \pi \int\limits_(a)^(b)f^2(x)\,dx\,.

If a curvilinear trapezoid were bounded both from below and from above by the curves y_1=f_1(x), y_2=f_2(x) , then

V= \pi \int\limits_(a)^(b)y_2^2\,dx- \pi \int\limits_(a)^(b)y_1^2\,dx= \pi\int\limits_(a )^(b)\Bigl(f_2^2(x)-f_1^2(x)\Bigr)dx\,.

Formula (3) can also be used to calculate the volume of a body of revolution in the case when the boundary of the rotating figure is given by parametric equations. In this case, one has to use the change of variable under the sign definite integral.

In some cases it turns out to be convenient to decompose bodies of revolution not into straight circular cylinders, but into figures of a different type.

For example, let's find the volume of the body obtained by rotating a curvilinear trapezoid around the y-axis. First, let's find the volume obtained by rotating a rectangle with a height of y#, at the base of which lies the segment . This volume is equal to the difference between the volumes of two straight circular cylinders

\Delta V_k= \pi y_k x_(k+1)^2- \pi y_k x_k^2= \pi y_k \bigl(x_(k+1)+x_k\bigr) \bigl(x_(k+1)- x_k\bigr).

But now it is clear that the desired volume is estimated from above and below as follows:

2\pi \sum_(k=0)^(n-1) m_kx_k\Delta x_k \leqslant V\leqslant 2\pi \sum_(k=0)^(n-1) M_kx_k\Delta x_k\,.

From this it easily follows formula for the volume of a body of revolution around the y-axis:

V=2\pi \int\limits_(a)^(b) xy\,dx\,.

Example 4 Find the volume of a ball of radius R.

Decision. Without loss of generality, we will consider a circle of radius R centered at the origin. This circle, rotating around the axis Ox, forms a ball. The circle equation is x^2+y^2=R^2 , so y^2=R^2-x^2 . Given the symmetry of the circle about the y-axis, we first find half of the desired volume

\frac(1)(2)V= \pi\int\limits_(0)^(R)y^2\,dx= \pi\int\limits_(0)^(R) (R^2-x^ 2)\,dx= \left.(\pi\!\left(R^2x- \frac(x^3)(3)\right))\right|_(0)^(R)= \pi\ !\left(R^3- \frac(R^3)(3)\right)= \frac(2)(3)\pi R^3.

Therefore, the volume of the entire sphere is \frac(4)(3)\pi R^3.

Example 5 Calculate the volume of a cone whose height is h and the radius of the base is r.

Decision. We choose a coordinate system so that the Ox axis coincides with the height h (Fig. 47), and we take the top of the cone as the origin. Then the equation of the line OA can be written as y=\frac(r)(h)\,x .

Using formula (3), we obtain:

V=\pi \int\limits_(0)^(h) y^2\,dx= \pi \int\limits_(0)^(h) \frac(r^2)(h^2)\,x ^2\,dx= \left.(\frac(\pi r^2)(h^2)\cdot \frac(x^3)(3))\right|_(0)^(h)= \ frac(\pi)(3)\,r^2h\,.

Example 6 Find the volume of the body obtained by rotating around the abscissa axis of the astroid \begin(cases)x=a\cos^3t\,\\ y=a\sin^3t\,.\end(cases)(Fig. 48).

Decision. Let's build an astroid. Consider half of the upper part of the astroid, located symmetrically about the y-axis. Using formula (3) and changing the variable under the definite integral sign, we find the limits of integration for the new variable t.

If x=a\cos^3t=0 , then t=\frac(\pi)(2) , and if x=a\cos^3t=a , then t=0 . Given that y^2=a^2\sin^6t and dx=-3a\cos^2t\sin(t)\,dt, we get:

V=\pi \int\limits_(a)^(b) y^2\,dx= \pi \int\limits_(\pi/2)^(0) a^2\sin^6t \bigl(-3a \cos^2t\sin(t)\bigr)\,dt= \ldots= \frac(16\pi)(105)\,a^3.

The volume of the entire body formed by the rotation of the astroid will be \frac(32\pi)(105)\,a^3.

Example 7 Find the volume of the body obtained by rotating around the ordinate axis of a curvilinear trapezoid bounded by the abscissa axis and the first arc of the cycloid \begin(cases)x=a(t-\sin(t)),\\ y=a(1-\cos(t)).\end(cases).

Decision. We use formula (4): V=2\pi \int\limits_(a)^(b)xy\,dx, and replace the variable under the integral sign, taking into account that the first arc of the cycloid is formed when the variable t changes from 0 to 2\pi . Thus,

\begin(aligned)V&= 2\pi \int\limits_(0)^(2\pi) a(t-\sin(t))a(1-\cos(t))a(1-\cos( t))\,dt= 2\pi a^3 \int\limits_(0)^(2\pi) (t-\sin(t))(1-\cos(t))^2\,dt= \\ &= 2\pi a^3 \int\limits_(0)^(2\pi)\bigl(t-\sin(t)- 2t\cos(t)+ 2\sin(t)\cos( t)+ t\cos^2t- \sin(t)\cos^2t\bigr)\,dt=\\ &= \left.(2\pi a^3\!\left(\frac(t^2 )(2)+ \cos(t)- 2t\sin(t)- 2\cos(t)+ \sin^2t+ \frac(t^2)(4)+ \frac(t)(4)\sin2t+ \frac(1)(8)\cos2t+ \frac(1)(3)\cos^3t\right))\right|_(0)^(2\pi)=\\ &= 2\pi a^3 \!\left(2\pi^2+1-2+\pi^2+\frac(1)(8)+ \frac(1)(3)-1+2- \frac(1)(8) - \frac(1)(3)\right)= 6\pi^3a^3. \end(aligned)

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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 to know the concept volumetric 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, in operating mathematical concepts and images, to cultivate 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 practice. "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.)

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

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

"Mathematical bunch".

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

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

Work in notebooks.

– The Newton-Leibniz formula was developed by the English physicist Isaac Newton (1643–1727) and 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.)

The year 1612 was very fruitful for the inhabitants of the Austrian city of Linz, where the then famous astronomer Johannes Kepler lived, especially for grapes. People were preparing wine barrels and wanted to know how to practically determine their volumes. (Slide 2)

- Thus, the considered works of Kepler marked the beginning of a whole stream of research, which culminated in the last quarter of the 17th century. design in the works of I. Newton and G.V. Leibniz differential and integral calculus. Since that time, the mathematics of magnitude variables has taken a leading place in the system of mathematical knowledge.

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

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

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

"Labyrinth".

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

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

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

Slide. “Instruction card” Calculation of volumes.

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

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

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

1. around the x-axis.

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

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

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

Consider an excerpt from famous fairy tale A. S. Pushkin “The Tale of Tsar Saltan, 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
In a barrel with son planted,
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 straight 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 group representatives 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.

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

2nd group. CONE (slide).

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

Exercise. Calculate the volume of the body obtained by rotation around the x-axis

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

I saw the picture. In this picture
There is a PYRAMID in the sandy desert.
Everything in the pyramid is extraordinary,
There is some mystery and mystery in it.
The Spasskaya Tower on Red Square
Both children and adults are well known.
Look at the tower - ordinary in appearance,
What's on top of her? Pyramid!

Exercise. Homework plot a function and calculate the volume of the pyramid

– Volumes various bodies we calculated 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).

Conclusion (slide).

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

Grading. (With commentary.)

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

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

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

In the formula, there must be a number before the integral. It just so happened - everything that spins in life is connected with this constant.

How to set the limits of integration "a" and "be", I think, 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 parabolic graph at the top. This is the function that is implied in the formula.

AT 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 integral is always non-negative , which is quite logical.

Calculate the volume of the 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 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 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 the axis of the figure bounded by lines,,

This is an example for independent solution. Complete Solution and the 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

Decision: Let's draw a flat figure in the drawing, bounded by lines ,,,, while not forgetting that the equation sets the axis:

The desired figure is shaded in blue. When it rotates around the axis, such a surreal donut with four corners is obtained.

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, a truncated cone is obtained. Denote the volume of this truncated cone by.

Consider the figure that is circled in green. If 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 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:

Now let's take a break and talk about geometric illusions.

People often have illusions associated with volumes, which Perelman (another) noticed in the book Interesting 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 in his entire life drinks a liquid with a volume of a room with an area of 18 square meters, which, on the contrary, seems to be too small.

In general, the education system in the USSR really was the best. The same book by Perelman, published back in 1950, develops very well, as the humorist said, reasoning and teaches you to look for original non-standard solutions to problems. Recently I re-read some chapters with great interest, I recommend it, it is accessible even for humanitarians. No, you don’t have to smile that I suggested a bespontovy pastime, erudition and a broad outlook in communication is a great thing.

After digression 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 a do-it-yourself example. Note that all things happen in the band , in other words, ready-made integration limits are actually given. Correctly draw graphs of trigonometric functions, I will remind you the material of the lesson about geometric transformations of graphs : if the argument is divisible by two: , then the graphs are stretched along the axis twice. It is desirable to find at least 3-4 points on trigonometric tables to 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.

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

In the formula, there must be a number before the integral. It just so happened - everything that spins in life is connected with this constant.

How to set the limits of integration "a" and "be", I think, 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. 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 quite logical.

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

This is a do-it-yourself example. 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

Decision: Let's depict in the drawing a flat figure bounded by lines , , , , while not forgetting that the equation defines the axis:

The desired figure is shaded in blue. When it rotates around the axis, such a surreal donut with four corners is obtained.

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, a truncated cone is obtained. Let's denote the volume of this truncated cone as .

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 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:

Now let's take a break and talk about geometric illusions.

People often have illusions associated with volumes, which Perelman (not the same) noticed in the book Interesting 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 in his entire life drinks a liquid with a volume of a room of 18 square meters, which, on the contrary, seems to be too small a volume.

In general, the education system in the USSR really was the best. The same book by Perelman, written by him back in 1950, develops very well, as the humorist said, reasoning and teaches you to look for original non-standard solutions to problems. Recently I re-read some chapters with great interest, I recommend it, it is accessible even for humanitarians. No, you don’t have to smile that I suggested a bespontovy pastime, erudition and a broad outlook in communication is 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 a do-it-yourself example. Please note that all things happen in the band, in other words, almost ready-made integration limits are given. Also try to draw the graphs correctly. trigonometric functions, if the argument is divisible by two: , then the graphs are stretched along the axis twice. Try to find at least 3-4 points according to trigonometric tables and make the drawing more accurate. 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 y-axis is also a fairly frequent guest in control work. In passing will be considered problem of finding the area of a figure the second way - integration along the axis, this will allow you not only to improve your skills, but also teach you how to find the most profitable solution. It also has a practical meaning! As my teacher of 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 manage our staff optimally.” 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 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 paragraph, first necessarily read the first one!

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

1) Let's execute the drawing:

It is easy to see that the function defines the upper branch of the parabola, and the function defines the lower branch of the parabola. Before us is a trivial parabola, which "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 considered in the lesson. Definite integral. How to calculate the area of a figure. Moreover, the area of \u200b\u200bthe figure is found as the sum of the areas:
- on the segment ;
- on the segment.

So:

What is wrong with the usual solution in this case? First, there are two integrals. Secondly, roots under integrals, and roots in integrals are not a gift, moreover, one can get confused in substituting the limits of integration. In fact, the integrals, of course, are not deadly, but in practice everything is much sadder, I just picked up “better” functions for the task.

There is a more rational solution: it consists in the transition to inverse functions and integration along the axis.

How to pass to inverse functions? Roughly speaking, you need to express "x" through "y". First, let's deal with the parabola:

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

With a straight line, everything is easier:

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. Moreover, on the segment, the straight line is located above the parabola, which means that the area of \u200b\u200bthe figure should be found using the formula already familiar to you: . What has changed in the formula? Only a letter, and nothing more.

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

Finding the area:

On the segment , therefore:

Pay attention to how I carried out the integration, this is the most rational way, and in the next paragraph of the assignment it will be clear why.

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

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

Answer:

2) Calculate the volume of the body formed by the rotation of this figure around the axis.

I will 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 the body of revolution, we will integrate along the axis. First we need to move on 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 the body of revolution should be found as the difference between the volumes.

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

We rotate the figure, circled in green, around the axis and denote it through the volume of the resulting body of revolution.

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:

How is it different from the formula of the previous paragraph? Only in letters.

And here's the advantage of integration that I was talking about a while ago, it's much easier to find than to preliminarily raise the integrand to the 4th power.

Answer:

However, a sickly butterfly.

Note that if the same flat figure is rotated around the axis, then a completely different body of revolution will turn out, of a different, naturally, volume.

Example 6

Given a flat figure bounded by lines , and an axis .

1) Go to inverse functions and find the area of a flat 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.

I. Volumes of bodies of revolution. Preliminarily study, according to the textbook by G. M. Fikhtengolts, Chapter XII, p ° p ° 197, 198 * Analyze in detail the examples given in p ° 198.

508. Calculate the volume of the body formed by the rotation of the ellipse Around the x-axis.

Thus,

530. Find the surface area, formed by rotation around the axis Ox of the arc of the sinusoid y \u003d sin x from the point X \u003d 0 to the point X \u003d It.

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

532. Calculate the surface area formed by

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

533. Calculate the area of the surface formed by the inversion of the loop of the curve 18 y-x(6-x)r around the x-axis.

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

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

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

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

from t = 0 to t = -.

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

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

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