The volume of a body of rotation around the axis is oh examples. How to calculate the volume of a body of revolution

Using integrals to find volumes of bodies of revolution

The practical usefulness of mathematics is due to the fact that without

specific mathematical knowledge, it is difficult to understand the principles of the device and use modern technology... Every person in their life has to do enough complex calculations, to use commonly used techniques, to find in reference books to apply the necessary formulas, to compose simple algorithms for solving problems. V modern society more and more specialties requiring high level education is associated with the direct application of mathematics. Thus, for a student, mathematics becomes a professionally significant subject. The leading role belongs to mathematics in the formation of algorithmic thinking, fosters the ability to act according to a given algorithm and design new algorithms.

Studying the topic of using an integral to calculate the volumes of bodies of revolution, I invite students to consider the topic: "Volumes of bodies of revolution using integrals" in optional classes. Below are guidelines for considering this topic:

1.Area of a flat figure.

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To find the volume of a body of revolution, formed by rotation curvilinear trapezoid around the axis Оx, bounded by the broken line y = f (x), axis Оx, straight lines x = a and x = b we calculate by the formula

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3. The volume of the cylinder.

https://pandia.ru/text/77/502/images/image011_58.gif "width =" 85 "height =" 51 "> .. gif" width = "13" height = "25"> .. jpg " width = "401" height = "355"> The taper is obtained by rotating right triangle ABC (C = 90) around the Ox axis on which the AC leg lies.

The segment AB lies on the straight line y = kx + c, where https://pandia.ru/text/77/502/images/image019_33.gif "width =" 59 "height =" 41 src = ">.

Let a = 0, b = H (H is the height of the cone), then Vhttps: //pandia.ru/text/77/502/images/image021_27.gif "width =" 13 "height =" 23 src = ">.

5. The volume of the truncated cone.

A truncated cone can be obtained by rotating a rectangular trapezoid ABCD (CDOx) around the Ox axis.

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

Since the straight line passes through point A (0; r).

Thus, the straight line looks like https://pandia.ru/text/77/502/images/image027_17.gif "width =" 303 "height =" 291 src = ">

Let a = 0, b = H (H is the height of the truncated cone), then https://pandia.ru/text/77/502/images/image030_16.gif "width =" 36 "height =" 17 src = "> = .

6. Volume of the sphere.

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

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

How to calculate the volume of a body of revolution
using a definite integral?

In general, there are a lot of interesting applications in integral calculus, with the help of a definite integral you can calculate the area of a figure, the volume of a body of revolution, the length of an arc, surface area of revolution, and much more. So it will be fun, please be optimistic!

Imagine some flat figure on the coordinate plane. Have you presented? ... I wonder who presented what ... =))) We have already found its area. But, in addition, this figure can also be rotated, and rotated in two ways:

- around the abscissa axis;
- around the ordinate axis.

This article will cover both cases. The second method of rotation is especially interesting, it causes the greatest difficulties, but in fact the solution is practically the same as in the more common rotation around the abscissa axis. As a bonus, I will return to the problem of finding the area of a figure, and I will tell you how to find the area in the second way - along the axis. It's not even so much a bonus as the material fits the theme well.

Let's start with the most popular spin type.

a flat figure around an axis

Calculate the volume of a solid obtained by rotating a shape bounded by lines around an axis.

Solution: As in the problem of finding the area, the solution starts with drawing a flat figure... That is, on a plane it is necessary to build a figure bounded by lines, and do not forget that the equation sets the axis. How to make a drawing more efficiently and faster, you can find out on the pages Graphs and Properties of Elementary Functions and . This is a Chinese reminder, and on this moment I don't stop anymore.

The drawing here is pretty simple:

The desired flat figure is shaded in blue, it is she who rotates around the axis. As a result of rotation, such a slightly ovoid flying saucer is obtained, which is symmetrical about the axis. In fact, the body has a mathematical name, but the reference book is too lazy to clarify something, so we go further.

How to calculate the volume of a body of revolution?

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

In the formula, there is always a number in front of the integral. It so happened - everything that revolves in life is connected with this constant.

How to set the limits of integration "a" and "bh", I think, is easy to guess from the completed drawing.

Function… what is this function? Let's take a look at the drawing. A flat figure is bounded by a parabola graph at the top. This is the function that is implied in the formula.

V practical assignments 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.

We calculate the volume of the body of revolution using this formula:

As I already noted, the integral is almost always 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 revolution 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 put into a flying saucer.

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

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

Consider two more challenging tasks, which are also often found in practice.

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

Solution: Draw 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 you rotate it around the axis, you get such a surreal donut with four corners.

The volume of the body of revolution is calculated as difference in body volumes.

First, let's look at the shape outlined in red. When it rotates around the axis, a truncated cone is obtained. Let us denote the volume of this truncated cone through.

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

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 shape circled in red is bounded from above by a straight line, therefore:

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

3) The volume of the sought-for 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 solution itself is often made shorter, something like this:

Now let's take some rest and talk about geometric illusions.

People often have illusions associated with volumes, which Perelman (another) noted 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.

After the lyrical digression, it is just appropriate to solve the creative task:

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 a do-it-yourself solution. Please note that all things take place in the strip, in other words, ready-made limits of integration are actually given. Draw graphs correctly trigonometric functions, I will remind 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 by trigonometric tables to more accurately complete the drawing. Complete solution and answer at the end of the tutorial. By the way, the task can be solved rationally and not very rationally.

Calculation of the volume of a body formed by rotation
a 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 frequent guest in control works... Along the way, it will be considered the problem of finding the area of a figure in 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. There is also a practical sense of life in this! As my teacher of the teaching methods of teaching mathematics recalled with a smile, many graduates thanked her with the words: "Your subject helped us a lot, now we are effective managers and manage the staff in an optimal way." Taking this opportunity, I also express my deep gratitude to her, especially since I use the knowledge gained on direct appointment =).

I recommend it to everyone, even complete teapots, for reading. Moreover, the assimilation of the material in the second section will be invaluable in calculating the double integrals.

You are given a flat figure bounded by lines,,.

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

Attention! Even if you only want to read the second paragraph, be sure to read the first one first!

Solution: 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 that "lies on its side."

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

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

That's why:

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

There is a more rational way of solving it: it consists in passing to inverse functions and integrating along the axis.

How do I go to reverse functions? Roughly speaking, you need to express "X" through "Y". Let's deal with the parabola first:

That's enough, but let's make sure that the same function can be pulled from the lower branch:

With a straight line, everything is easier:

Now let's look at the axis: please, periodically tilt your head to the right 90 degrees as you explain (this is not a joke!). The shape we need lies on the segment 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 you are already familiar with: ... 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!

Find 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 have doubts about the correctness of the integration, I will find the derivatives:

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

Answer:

2) Let's 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 shape 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 revolution, we will integrate along the axis. First you need to go to the inverse functions. This has already been done and detailed in the previous paragraph.

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

Rotate the shape outlined in red around the axis, resulting in a truncated cone. Let's designate this volume through.

Rotate the shape, 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:

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

And here is the integration advantage I talked about recently, much easier to find than to first raise the integrand to the 4th power.

Answer:

Note that if you rotate the same flat figure around the axis, you get a completely different body of rotation, of a different volume, of course.

You are given a flat figure bounded by lines and an axis.

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

This is an example for a do-it-yourself solution. Those interested can also find the area of the figure in the "usual" way, thereby checking point 1). But if, I repeat, you rotate a flat figure around an 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).

The complete solution of the two proposed points of the assignment at the end of the lesson.

Oh, and don't forget to tilt your head to the right to understand the bodies of revolution and within the integration!

I wanted, it was already, to finish the article, but today they brought interesting example just for finding the volume of a body of revolution around the ordinate axis. Freshly:

Calculate the volume of the body formed by rotation around the axis of the figure bounded by the curves and.

Solution: Let's execute the drawing:

Along the way, we get acquainted with the graphs of some other functions. Such is an interesting graph. even function ….

I. Volumes of bodies of revolution. Preliminarily study, according to the textbook of G.M.Fikhtengolts, chapter XII, n ° n ° 197, 198 * Disassemble in detail the examples given in n ° 198.

508. Calculate the volume of the body formed by the rotation of the ellipse around the Ox axis.

Thus,

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

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

532. Calculate the area of the surface formed

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

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

534. Find the surface of the torus produced by the rotation of the circle X2 - j - (y - Z) 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 area of the surface formed by the rotation of the loop of the curve x = 9t2, y = St - 9t3 around the Ox axis.

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 Ox axis

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 cardioidAround the polar axis.

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

Additional tasks for chapter IV

Areas of flat figures

541. The entire area of the area bounded by the curve And the Oh axis.

542. Find the area of the area bounded by the curve

And the Oh axis.

543. Find the part of the area of the area located in the first quadrant and bounded by the curve

l coordinate axes.

544. Find the area of the area contained inside

loops:

545. Find the area of the region bounded by one loop of the curve:

546. Find the area of the area contained within the loop:

547. Find the area of the area bounded by the curve

And the Oh axis.

548. Find the area of the area bounded by the curve

And the Oh axis.

549. Find the area of the area bounded by the Oxr axis

straight and curve

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

Let us prove that it is a body of revolution is cubic 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, let us prove that this body of revolution is regular if we choose the plane Oyz as \ Pi, which is perpendicular to the axis of rotation. 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). Further, if S (x_1) \ leqslant S (x_2) then it means that. But the projections of the sections on 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 a circle of radius f (x_1) is contained in a circle of radius f (x_2).

So, the body of rotation is regular. Therefore, it is cubic 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 the curvilinear trapezoid was 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 a rotating figure is specified by parametric equations. In this case, one has to use the change of variable under the definite integral sign.

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 kind.

For example, let's find volume of a body obtained by rotating a curved trapezoid around the ordinate axis... First, we find the volume obtained by rotating a rectangle with height y #, at the base of which the segment lies. 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 required 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 \ ,.

It easily follows from this formula for the volume of a body of revolution around the ordinate axis:

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

Example 4. Let us find the volume of a ball of radius R.

Solution. Without loss of generality, we will consider a circle of radius R centered at the origin. This circle, rotating around the Ox axis, forms a ball. The equation of the circle is x ^ 2 + y ^ 2 = R ^ 2, so y ^ 2 = R ^ 2-x ^ 2. Taking into account the symmetry of the circle about the ordinate axis, we first find half of the required 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 ball 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 r.

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

Using formula (3), we get:

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. Let us find the volume of the body obtained by rotation around the abscissa axis of the astroid \ begin (cases) x = a \ cos ^ 3t \, \\ y = a \ sin ^ 3t \,. \ end (cases)(fig. 48).

Solution. Let's build an astroid. Consider half of the upper part of the astroid, located symmetrically about the ordinate 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. Considering 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 whole body formed by the rotation of the astroid will be \ frac (32 \ pi) (105) \, a ^ 3.

Example 7. Let us find the volume of the body obtained by rotating a curvilinear trapezoid around the ordinate axis 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).

Solution. Let's 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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As with the problem of finding the area, you need confident drawing skills - this is almost the most important thing (since the integrals themselves will often be easy). Master the literate and fast technique graphing can be done using teaching materials and Geometric transformations of graphs. But, in fact, I have already repeatedly spoken about the importance of drawings in the lesson.

In general, there are a lot of interesting applications in integral calculus, using a definite integral you can calculate the area of a figure, the volume of a body of revolution, the length of an arc, the area of a surface of revolution, and much more. So it will be fun, please be optimistic!

- around the abscissa axis;
- around the ordinate axis.

Let's start with the most popular spin type.

a flat figure around an axis

Example 1

Calculate the volume of a solid obtained by rotating a shape bounded by lines around an axis.

Solution: As in the problem of finding the area, the solution starts with drawing a flat figure... That is, on a plane it is necessary to build a figure bounded by lines, and do not forget that the equation sets the axis. How to make a drawing more efficiently and faster, you can find out on the pages Graphs and Properties of Elementary Functions and Definite integral. How to calculate the area of a shape... This is a Chinese reminder, and I don't stop at this point anymore.

The drawing here is pretty simple:

How to calculate the volume of a body of revolution?

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

A number must always be present in the formula before the integral. It so happened - everything that revolves in life is connected with this constant.

How to set the limits of integration "a" and "bh", I think, is easy to guess from the completed drawing.

Function… what is this function? Let's take a look at the drawing. A flat figure is bounded by a parabola graph at the top. This is the function that is implied in the formula.

In practical exercises, 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.

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

As I already noted, the integral is almost always 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 lines,

This is an example for a do-it-yourself solution. Complete solution and answer at the end of the tutorial.

Consider two more complex tasks that are also common in practice.

Example 3

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

Solution: Draw 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 you rotate it around the axis, you get such a surreal donut with four corners.

The volume of the body of revolution is calculated as difference in body volumes.

First, let's look at the shape outlined in red. When it rotates around the axis, a truncated cone is obtained. Let us denote the volume of this truncated cone through.

Consider the shape outlined in green. If you rotate this figure around the axis, you will also get a truncated cone, only slightly smaller. Let's denote its volume through.

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 shape circled in red is bounded from above by a straight line, therefore:

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

3) The volume of the sought-for 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 solution itself is often made shorter, something like this:

Now let's take some rest and talk about geometric illusions.

In general, the education system in the USSR was indeed the best. The same book by Perelman, published back in 1950, very well develops, as the humorist said, reasoning and teaches to look for original non-standard solutions problems. Recently I re-read some chapters with great interest, I recommend it, it is available even for the humanities. No, there is no need to smile that I offered a free time, erudition and a broad outlook in communication is a great thing.

After the lyrical digression, it is just appropriate to solve the 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 a do-it-yourself solution. Please note that all things take place in the strip, 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 divisible by two:, then the graphs are stretched along the axis twice. It is desirable to find at least 3-4 points by trigonometric tables to more accurately complete the drawing. Complete solution and answer at the end of the tutorial. By the way, the task can be solved rationally and not very rationally.

Calculation of the volume of a body formed by rotation
a 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 frequent guest in tests. Along the way, it will be considered the 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. There is also a practical sense of life in this! As my teacher of the teaching methods of teaching mathematics recalled with a smile, many graduates thanked her with the words: "Your subject helped us a lot, now we are effective managers and manage the staff in an optimal way." Taking this opportunity, I also express my deep gratitude to her, especially since I use the acquired knowledge for its intended purpose =).

I recommend it to everyone, even complete teapots, for reading. Moreover, the assimilation of the material in the second section will be invaluable in calculating the double integrals.

Example 5

You are given a flat figure bounded by lines,,.

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

Attention! Even if you only want to read the second paragraph, first necessarily read the first one!

Solution: 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 that "lies on its side."

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

That's why:

There is a more rational way of solving it: it consists in passing to inverse functions and integrating along the axis.

How do I go to reverse functions? Roughly speaking, you need to express "X" through "Y". Let's deal with the parabola first:

That's enough, but let's make sure that the same function can be pulled from the lower branch:

With a straight line, everything is easier:

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

Find 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 point of the assignment it will be clear why.

For readers who have doubts about the correctness of the integration, I will find the derivatives:

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

Answer:

2) Let's 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 shape 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 revolution, we will integrate along the axis. First you need to go to the inverse functions. This has already been done and detailed in the previous paragraph.

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

Rotate the shape outlined in red around the axis, resulting in a truncated cone. Let's designate this volume through.

Rotate the shape, 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:

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

And here is the integration advantage I talked about recently, much easier to find than to first raise the integrand to the 4th power.

Answer:

However, a sickly butterfly.

Note that if you rotate the same flat figure around the axis, you get a completely different body of rotation, of a different volume, of course.

Example 6

You are given a flat figure bounded by lines and an axis.

The complete solution of the two proposed points of the assignment at the end of the lesson.

Oh, and don't forget to tilt your head to the right to understand the bodies of revolution and within the integration!

The volume of a body of rotation around the axis is oh examples. How to calculate the volume of a body of revolution

How to calculate the volume of a body of revolution using a definite integral?

a flat figure around an axis

How to calculate the volume of a body of revolution?

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

a flat figure around an axis

How to calculate the volume of a body of revolution?

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

How to calculate the volume of a body of revolution
using a definite integral?

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

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