home » Haircuts » How to calculate the volume of a body of revolution using a definite integral? Calculation of the volume of a body formed by rotation.

How to calculate the volume of a body of revolution using a definite integral? Calculation of the volume of a body formed by rotation.

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

Besides finding the area of a plane figure using a definite integral 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 . 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 literate and fast technology plotting can be done using methodological material . But, in fact, I have already talked about the importance of drawings several times in class. .

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 a body and much more. So it will be fun, please stay optimistic!

Imagine some flat figure on the coordinate plane. Introduced? ... 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 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 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.

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 it is necessary to construct a figure bounded by lines, and do not forget that the equation defines 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 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 look 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 using the formula:

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

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

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

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

Let's calculate the volume of a body of rotation 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 green. If you rotate this figure around the axis, you will also get a truncated cone, only a little smaller. Let's denote its volume by.

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

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

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

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

3) Volume of the desired body of rotation:

Answer:

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

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

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

People often have illusions associated with volumes, which were noticed by Perelman (not that one) in the book Entertaining geometry. Look at the flat figure in the solved problem - it seems to be small in area, and the volume of the body of revolution is just over 50 cubic units, which seems too large. By the way, the average person drinks the equivalent of a room of 18 square meters of liquid in his entire life, which, on the contrary, seems too small a volume.

In general, the education system in the USSR was truly the best. The same book by Perelman, written by him back in 1950, very well develops, as the humorist said, thinking and teaches one to look for original, non-standard solutions to problems. I recently re-read some of the chapters with great interest, I recommend it, it’s accessible even for humanists. No, you don’t need to smile that I offered a free time, erudition and broad horizons in communication are a great thing.

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

Example 4

Calculate the volume of a body formed by rotation about the axis of a flat figure bounded by lines,, where.

This is an example for you to solve on your own. Please note that all things happen in the band, in other words, practically ready-made limits of integration are given. Also try to draw the graphs correctly. trigonometric functions, if the argument is divided by two:, then the graphs are stretched along the axis twice. 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 rotating 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 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 point in this. life meaning! 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 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 in moving to inverse functions and integration 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. Moreover, 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 setstrictly 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 by 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.

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.

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 charting techniques using teaching materials and Geometric Transformations of Graphs. But, in fact, I have already talked about the importance of drawings several times in class.

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 stay optimistic!

– 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 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

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 on at this moment I don't stop anymore.

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 using the formula:

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

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

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

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

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

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

Answer:

Example 2

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

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

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

Example 3

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

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

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

Let us calculate the volume of the body of 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 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:

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

People often have illusions associated with volumes, which was noticed by Perelman (another) in the book Entertaining geometry. Look at the flat figure in the solved problem - it seems to be small in area, and the volume of the body of revolution is just over 50 cubic units, which seems too large. By the way, the average person drinks the equivalent of a room of 18 square meters of liquid in his entire life, which, on the contrary, seems too small a volume.

In general, the education system in the USSR was truly the best. The same book by Perelman, published back in 1950, very well develops, as the humorist said, understanding and teaches you to look for original non-standard solutions 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 cases occur in the band, in other words, ready-made limits of integration are actually given. Draw the graphs of trigonometric functions correctly, let me remind you of the lesson material about geometric transformations of graphs: if the argument is divided by two: , then the graphs are stretched twice along the axis. It is advisable to find at least 3-4 points according to trigonometric tables to complete the drawing more accurately. Full solution and answer at the end of the lesson. By the way, the task can be solved rationally and not very rationally.

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 frequent guest in tests. 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 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:

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:

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 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:

However, not a sickly butterfly.

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

Example 6

Given a flat figure bounded by lines and an axis.

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

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

A 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!

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.

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

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

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

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

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

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

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

Answer:

Example 2

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

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

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

Example 3

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

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