Except finding the area of a plane figure using a definite integral (see 7.2.3.) the most important application of the topic is calculating the volume of a body of rotation. The material is simple, but the reader must be prepared: you must be able to solve indefinite integrals medium complexity and apply the Newton-Leibniz formula in definite integral, n You also need strong drawing skills. 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 rotation, the length of an arc, the surface area of a body and much more. Imagine some flat figure on the coordinate plane. Introduced? ... Now this figure can also be rotated, and rotated in two ways:
– around the x-axis ;
– around the ordinate axis .
Let's look at both cases. The second method of rotation is especially interesting; it causes the most 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 body volume, formed by rotation flat figure around an axis OX
Example 1
Calculate the volume of a body obtained by rotating a figure bounded by lines around an axis.
Solution: As in the problem of finding the area, the solution begins with a drawing of a flat figure. That is, on a plane XOY it is necessary to construct a figure bounded by the lines , and do not forget that the equation specifies the axis. The drawing here is quite simple:
The desired flat figure is shaded in blue; it is the one that rotates around the axis. As a result of rotation, the result is a slightly ovoid flying saucer with two sharp peaks on the axis 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 rotation? If a body is formed as a result of rotation around an axisOX, it is mentally divided into parallel layers of small thickness dx, which are perpendicular to the axis OX. The volume of the entire 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 in height dx and with base radius f(x). Then the volume of one layer is the product of the base area π f 2 per cylinder height ( dx), or π∙ f 2 (x)∙dx. And the area of the entire body of rotation is the sum of elementary volumes, or the corresponding definite integral. The volume of a body of revolution can be calculated using the formula:
.
How to set the limits of integration “a” and “be” can be easily guessed 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 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 very logical. Let's calculate the volume of a body of rotation 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 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 this is 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 an axis OX 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.
Example 3
Calculate the volume of the body obtained by rotating the figure bounded by the lines , , and around the abscissa axis.
Solution: Let us depict in the drawing a flat figure bounded by the lines , , , , without forgetting that the equation x= 0 specifies the axis OY:
The desired figure is shaded in blue. When it rotates around an axis OX the result is a flat, angular donut (a washer with two conical surfaces).
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 OX the result is a truncated cone. Let us denote the volume of this truncated cone by V 1 .
Consider the figure that is circled green. If you rotate this figure around the axis OX, then you get the same truncated cone, only a little smaller. Let us denote its volume by V 2 .
It is obvious that the difference in volumes V = V 1 - V 2 is 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:
flat figure around an axis
Example 3
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 “normal” way. Moreover, the area of the figure is found as the sum of the areas:
- on the segment 
;
- on the segment.
That's why:
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 : Axis integration limits should be placedstrictly 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 
, rather than first raising the integrand to the 4th power.
Answer: 
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 7
Calculate the volume of a body formed by rotation around the axis of a figure bounded by curves and .
Solution: Let's make a drawing:
Along the way, we get acquainted with the graphs of some other functions. Here is an interesting graph of an even function...
For the purpose of finding the volume of a body of revolution, it is enough to use the right half of the figure, which I shaded in blue. Both functions are even, their graphs are symmetrical about the axis, and our figure is symmetrical. Thus, the shaded right part, rotating around the axis, will certainly coincide with the left unshaded part.
I. Volumes of bodies of revolution. Preliminarily study Chapter XII, paragraphs 197, 198 from the textbook by G. M. Fikhtengolts * Analyze in detail the examples given in paragraph 198.
508. Calculate the volume of a body formed by rotating an ellipse around the Ox axis.
Thus,
530. Find the surface area formed by rotation around the Ox axis of the sinusoid arc y = sin x from point X = 0 to point X = It.
531. Calculate the surface area of a cone with height h and radius r.
532. Calculate the surface area formed
rotation of the astroid x3 -)- y* - a3 around the Ox axis.
533. Calculate the surface area formed by rotating the loop of the curve 18 ug - x (6 - x) z around the Ox axis.
534. Find the surface of the torus produced by the rotation of the circle X2 - j - (y-3)2 = 4 around the Ox axis.
535. Calculate the surface area formed by the rotation of the circle X = a cost, y = asint around the Ox axis.
536. Calculate the surface area formed by the rotation of the loop of the curve x = 9t2, y = St - 9t3 around the Ox axis.
537. Find the surface area formed by rotating the arc of the curve x = e*sint, y = el cost around the Ox axis
from t = 0 to t = —.
538. Show that the surface produced by the rotation of the cycloid arc x = a (q> -sin φ), y = a (I - cos φ) around the Oy axis is equal to 16 u2 o2.
539. Find the surface obtained by rotating the cardioid around the polar axis.
540. Find the surface area formed by the rotation of the lemniscate 
Around the polar axis.
Additional tasks for Chapter IV
Areas of plane figures
541. Find the entire area of the region bounded by the curve 
And the axis Ox.
542. Find the area of the region bounded by the curve
And the axis Ox.
543. Find the part of the area of the region located in the first quadrant and bounded by the curve
l coordinate axes.
544. Find the area of the region contained inside
loops: 
545. Find the area of the region bounded by one loop of the curve:
546. Find the area of the region contained inside the loop: 
547. Find the area of the region bounded by the curve
And the axis Ox.
548. Find the area of the region bounded by the curve
And the axis Ox.
549. Find the area of the region bounded by the Oxr axis
straight and curve 
Topic: “Calculating 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 identify curved trapezoids from a series geometric shapes and practice 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 development logical thinking, competent mathematical speech, accuracy when constructing drawings;
to cultivate interest in the subject, in operating with mathematical concepts and images, to cultivate will, independence, and perseverance in achieving the final result.
During the classes
I. Organizational moment.
Greetings from the group. Communicate lesson objectives to students.
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 thinks: “If the living one says, I’ll kill her; if the dead one says, I’ll release her.” 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 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 “Exclude superfluous word”.
(Students say an 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.
Exercise. Recover the gaps. (The student comes out and writes in the required words with a marker.)
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 construct 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? (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 volumetric, but how to 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 justified they were is another matter.
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.
Thus, the considered works of Kepler marked the beginning of a whole stream of research that culminated in the last quarter of the 17th century. design in the works of I. Newton and G.V. Leibniz of differential and integral calculus. From that time on, the mathematics of variables took a leading place in the system of mathematical knowledge.
Today we will do this practical activities, hence,
The topic of our lesson: “Calculating the volumes of bodies of rotation using a definite integral.”
You will learn the definition of a body of revolution by completing the following task.
“Labyrinth”.
Exercise. 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 particular body, in particular, a body of revolution.
A body of revolution is a body obtained by rotation curved trapezoid around its base (Fig. 1, 2)
The volume of a body of revolution 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.
Students write down basic formulas in a notebook.
The teacher explains the solutions to the examples on the board.
1. Find the volume of the body obtained by rotating around the ordinate 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 x-axis y = , x = 4, y = 0.
Solution.
V. Math simulator.
2. The set of all antiderivatives of a given function is called
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. Consolidating new material
Calculate the volume of the body formed by the rotation of the petal around the x-axis y = x2, y2 = x.
Let's build graphs of the function. y = x2, y2 = x. Let's transform the graph y2 = x to the form y = .
We have V = V1 - V2 Let’s calculate the volume of each function:
Conclusion:
Definite integral- this is some 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.
Development modern science is unthinkable without using the integral. In this regard, it is necessary to begin studying it within the framework of secondary specialized education!
VI. Grading.(With commentary.)
The great Omar Khayyam - mathematician, poet, philosopher. He encourages us to be masters of our own destiny. Let's listen to an excerpt from his work:
You say, this life is one moment.
Appreciate it, draw inspiration from it.
As you spend it, so it will pass.
Don't forget: she is your creation.
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 may intersect the figure if it is the axis of symmetry of the figure.
Theorem 2.
, axis 
and straight segments 
And 
rotates around an axis 
. Then the volume of the resulting body of rotation can be calculated using the formula
(2)
Proof.
For such a body, the cross section with abscissa 
is a circle of radius 
, Means 
and formula (1) gives the required result.
If the figure is limited by the graphs of two continuous functions 
And 
, and line segments 
And 
, and 
And 
, then upon rotation around the x-axis we obtain a body whose volume
Example 3.
Calculate the volume of a torus obtained by rotating a circle bounded by a circle 
around the abscissa axis.
R 
decision.
The indicated circle is limited below by the graph of the function 
, and from above – 
. The difference of the squares of these functions:
Required 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 
, rotates around the base. Calculate the volume of the resulting body (“lemon” by Cavalieri).
R 
decision.
We will place the parabola as shown in the figure. Then its equation 
, and 
. Let's find the value of the parameter 
:
. So, the required volume:
Theorem 3.
Let a curvilinear trapezoid bounded by the graph of a continuous non-negative function 
, axis 
and straight segments 
And 
, and 
, rotates around an axis 
. Then the volume of the resulting body of rotation can be found by the formula
(3)
The idea of proof.
We split the segment 
dots 
, into parts and draw straight lines 
. The entire trapezoid will be decomposed into strips, which can be considered approximately rectangles with a base 
and height 
.
We cut the resulting cylinder by rotating such a rectangle along its generatrix and unfold it. We get an “almost” parallelepiped with dimensions: 
,
And 
. Its volume 
. So, for the volume of a body of revolution we will have the approximate equality
To obtain exact equality, one must go 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.
Note 1.
In Theorems 2 and 3 the condition 
can be omitted: formula (2) is generally insensitive to the sign 
, and in formula (3) it is sufficient 
replaced by 
.
Example 5.
Parabolic segment (base 
, height 
) rotates around the height. Find the volume of the resulting body.
Solution.
Let's place the parabola as shown in the figure. And although the axis of rotation intersects the figure, it - the axis - is the axis of symmetry. Therefore, we need to consider only the right half of the segment. Parabola equation 
, and 
, Means 
. For volume we have:
Note 2.
If the curvilinear boundary of a curvilinear trapezoid is given by parametric equations 
,
,
And 
,
then you can use formulas (2) and (3) with the replacement 
on 
And 
on 
when it changes t from 
before 
.
Example 6.
The figure is limited by the first arc of the cycloid 
,
,
, and the x-axis. Find the volume of the body obtained by rotating this figure around: 1) axis 
; 2) axes 
.
Solution.
1) General formula 
In our case:
2) General formula 
For our figure:
We invite students to carry out all the calculations themselves.
Note 3.
Let a curved sector bounded by a continuous line 
and rays 
,
, rotates around a polar axis. The volume of the resulting body can be calculated using the formula.
Example 7.
Part of a figure bounded by a cardioid 
, lying outside the circle 
, rotates around a polar axis. Find the volume of the resulting body.
Solution.
Both lines, and therefore the figure they limit, are symmetrical about the polar axis. Therefore, it is necessary to consider only that part for which 
. The curves intersect at 
And
at 
. Further, the figure can be considered as the difference of two sectors, and therefore the volume can be calculated as the difference of two integrals. We have:
Tasks for an independent decision.
1. A circular segment whose base 
, height 
, rotates 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 abscissa axis. Find the volume of the resulting body.
4. Figure bounded by lines 
And 
rotates around the x-axis. Find the volume of the body of revolution.