Calculation of the volume of a body of rotation using a definite integral. Volume of a body of revolution

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

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; 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 ​​rotation and much more. So it will be fun, please be optimistic!

Imagine some flat figure on the coordinate plane. Represented? ... 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 x-axis;
- around the y-axis.

This article will examine 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. As a bonus I will return to problem of finding the area of ​​a figure, and I’ll tell you how to find the area in the second way - along the axis. It’s not so much a bonus as the material fits well into the topic.

Let's start with the most popular type of rotation.

flat figure around an axis

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 the plane it is necessary to construct a figure bounded by the lines , and do not forget that the equation specifies the axis. How to complete a drawing more efficiently and quickly can be found on the pages Graphs and properties of Elementary functions And . This is a Chinese reminder, and at this point I will not dwell further.

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 the rotation, the result is a slightly ovoid flying saucer that is symmetrical about the axis. In fact, the body has a mathematical name, but I’m too lazy to clarify anything in the reference book, so we move on.

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, the number must be present before the integral. It so happened - everything that spins 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 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 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 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.

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.

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 revolution 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 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 was noticed by Perelman (another) 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 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.

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

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 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 divisible by two: , then the graphs are stretched along the axis twice. It is desirable to find at least 3-4 points according to 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.

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 =).

I recommend it to everyone, even complete dummies. Moreover, the material learned in the second paragraph will provide invaluable assistance in calculating double integrals.

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, be sure to read the first one first!

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, there are roots under integrals, 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 limits of integration 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 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 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:

How is it different from the formula of 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:

Please 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.

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:

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...

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 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:

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 revolution 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 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 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 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 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 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:

How is it different from the formula of 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:

However, not a sickly butterfly.

Please 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.

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 revolution. 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 ​​\u200b\u200ba figure, the volume of a body of revolution, the length of an arc, the surface area of ​​\u200b\u200bthe body, and much more. Imagine some flat figure on the coordinate plane. Represented? ... Now this figure can also be rotated, and rotated in two ways:

– around the x-axis ;

– around the ordinate axis .

Let's look at both cases. The second method of rotation is especially interesting, it causes the greatest difficulties, but in fact the solution is almost the same as in the more common rotation around the x-axis. Let's start with the most popular type of rotation.

Calculation of the volume of a body formed by rotating a 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 the plane XOY it is necessary to construct a figure bounded by lines , while not forgetting that the equation defines the axis . The drawing here is pretty simple:

The desired flat figure is shaded in blue; it is 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 revolution? If the body is formed as a result of rotation around an axisOX, it is mentally divided into parallel layers of small thickness dx that are perpendicular to the axis OX. The volume of the 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 high dx and with base radius f(x). Then the volume of one layer is the product of the base area π f 2 to the height of the cylinder ( dx), or π∙ f 2 (x)∙dx. And the area of ​​the entire body of revolution is the sum of elementary volumes, or the corresponding definite integral. The volume of a body of revolution can be calculated using the formula:

.

How to set the integration limits "a" and "be" is easy to guess from the completed drawing. Function... what is this function? Let's look at the drawing. The flat figure is bounded by the parabola graph from above. This is the function that is implied in the formula. In practical tasks, a flat figure can sometimes be located below the axis OX. This does not change anything - the function in the formula is squared: f 2 (x), Thus, the volume of a body of revolution is always non-negative, which is quite logical. Calculate the volume of the body of revolution using this formula:

.

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

Answer:

In 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 it is the most universal formulation. There may be cubic centimeters, there may be cubic meters, there may be cubic kilometers, etc., that's how many little green men your imagination can fit into a flying saucer.

Example 2

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

This is an example for 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 around the abscissa axis of the figure bounded by the lines , , and .

Solution: Let us depict in the drawing a flat figure bounded by lines , , , , while not forgetting that the equation x= 0 specifies the axis OY:

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

Let us calculate the volume of the body of revolution as difference in volumes of bodies. First, let's look at the figure circled in red. When it rotates around the 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 in 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 for finding 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 rotation:

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:

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). You can master competent and fast graphing techniques with the help of teaching materials and Geometric transformations of graphs. But, in fact, I have 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 rotation, arc length, surface area of ​​rotation, and much more. So it will be fun, please be optimistic!

Imagine some flat figure on the coordinate plane. Represented? ... 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 examine 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. As a bonus I will return to problem of finding the area of ​​a figure, and tell you how to find the area in the second way - along the axis. It’s not so much a bonus as the material fits well into the topic.

Let's start with the most popular type of rotation.

flat figure around an axis

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 the plane it is necessary to construct a figure bounded by the lines , and do not forget that the equation specifies the axis. How to complete a drawing more efficiently and quickly can be found on the pages Graphs and properties of Elementary functions And Definite integral. How to calculate the area of ​​a figure. This is a Chinese reminder, and at this point I will not dwell further.

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 the rotation, the result is a slightly ovoid flying saucer that is symmetrical about the axis. In fact, the body has a mathematical name, but I’m too lazy to clarify anything in the reference book, so we move on.

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, 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 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 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 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 revolution 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 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 was noticed by Perelman (another) 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 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, 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 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 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 divisible by two: , then the graphs are stretched along the axis twice. It is desirable to find at least 3-4 points according to 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.

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 =).

I recommend it to everyone, even complete dummies. Moreover, the material learned in the second paragraph will provide invaluable assistance in calculating double integrals.

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 limits of integration 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 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 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:

How is it different from the formula of 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:

However, not a sickly butterfly.

Please 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!