The volume of a body of revolution around y. Lesson "Calculating the volumes of bodies of revolution using a 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). You can master a competent and fast graphing technique with the help of methodological 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 \u200b\u200ba figure, the volume of a body of revolution, 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 y-axis.

In this article, both cases will be discussed. 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. As a bonus, I will return to the problem of finding the area of a figure, and tell you how to find the area in the second way - along the axis. Not even so much a bonus as the material fits well into the theme.

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

flat figure around an axis

Example 1

Calculate the volume of the body obtained by rotating the figure bounded by lines around the axis.

Solution: As in the area problem, the solution starts with a drawing of a flat figure. That is, on the plane it is necessary to build a figure bounded by lines , , while not forgetting that the equation defines the axis . How to make a drawing more rationally and faster 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 I don't stop at this point.

The drawing here is pretty simple:

The desired flat figure is shaded in blue, and it is it that rotates around the axis. As a result of rotation, such a slightly egg-shaped flying saucer is obtained, which is symmetrical about the axis. In fact, the body has a mathematical name, but it’s too lazy to specify something 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, there must be a number before the integral. It just so happened - everything that spins in life is connected with this constant.

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

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

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

Calculate the volume of the body of revolution using this formula:

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

Answer:

In the answer, it is necessary to indicate the dimension - cubic units. That is, in our body of rotation there are approximately 3.35 "cubes". Why exactly cubic units? Because the most universal formulation. There may be cubic centimeters, there may be cubic meters, there may be cubic kilometers, etc., that's how many little green men your imagination can fit into a flying saucer.

Example 2

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

This is a do-it-yourself example. Full solution and answer at the end of the lesson.

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

Example 3

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

Solution: Draw a flat figure in the drawing, bounded by lines , , , , while not forgetting that the equation defines the axis:

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

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

First, let's look at the figure that is circled in red. When it rotates around the axis, a truncated cone is obtained. Let's denote the volume of this truncated cone as .

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

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

We use the standard formula for finding the volume of a body of revolution:

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

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

3) The volume of the desired body of revolution:

Answer:

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

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

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

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

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

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

Example 4

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

This is a do-it-yourself example. Note that all things happen in the band , in other words, ready-made integration limits are actually given. Correctly draw graphs of trigonometric functions, I will remind you the material of the lesson about geometric transformations of graphs: if the argument is divisible by two: , then the graphs are stretched along the axis twice. It is desirable to find at least 3-4 points 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 y-axis is also a fairly frequent visitor in tests. In passing will be considered problem of finding the area of a figure the second way - integration along the axis, this will allow you not only to improve your skills, but also teach you how to find the most profitable solution. It also has a practical meaning! As my teacher of mathematics teaching methods recalled with a smile, many graduates thanked her with the words: “Your subject helped us a lot, now we are effective managers and manage our staff optimally.” Taking this opportunity, I also express my great gratitude to her, especially since I use the acquired knowledge for its intended purpose =).

I recommend it for everyone to read, even complete dummies. Moreover, the assimilated material of the second paragraph will be of invaluable help 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 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, which "lies on its side."

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

How to find the area of a figure? It can be found in the "usual" way, which was considered in the lesson. Definite integral. How to calculate the area of a figure. Moreover, the area of \u200b\u200bthe figure is found as the sum of the areas:
- on the segment ;
- on the segment.

So:

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

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

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

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

With a straight line, everything is easier:

Now look at the axis: please periodically tilt your head to the right 90 degrees as you explain (this is not a joke!). The figure we need lies on the segment, which is indicated by the red dotted line. Moreover, on the segment, the straight line is located above the parabola, which means that the area of \u200b\u200bthe figure should be found using the formula already familiar to you: . What has changed in the formula? Only a letter, and nothing more.

! Note: Integration limits along the axis should be set strictly from bottom to top!

Finding the area:

On the segment , therefore:

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

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

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

Answer:

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

I will redraw the drawing in a slightly different design:

So, the figure shaded in blue rotates around the axis. The result is a "hovering butterfly" that rotates around its axis.

To find the volume of the body of revolution, we will integrate along the axis. First we need to move on to inverse functions. This has already been done and described in detail in the previous paragraph.

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

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

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

The volume of our butterfly is equal to the difference in volumes.

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

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

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

Answer:

However, a sickly butterfly.

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

Example 6

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

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

This is a do-it-yourself example. Those who wish can also find the area of \u200b\u200bthe figure in the "usual" way, thereby completing the test of point 1). But if, I repeat, you rotate a flat figure around the axis, then you 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 items of the task at the end of the lesson.

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

flat figure around an axis

Example 3

Given a flat figure bounded by lines , , .

1) Find the area of a flat figure bounded by these lines.

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

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

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, which "lies on its side."

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

How to find the area of a figure? It can be found in the "normal" way. Moreover, the area of \u200b\u200bthe figure is found as the sum of the areas:

- on the segment ;

- on the segment.

So:

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

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

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

With a straight line, everything is easier:

! Note : Axis integration limits should be arrangedstrictly from bottom to top !

Finding the area:

On the segment , therefore:

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

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

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

Answer:

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

I will redraw the drawing in a slightly different design:

So, the figure shaded in blue rotates around the axis. The result is a "hovering butterfly" that rotates around its axis.

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

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

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

The volume of our butterfly is equal to the difference in volumes.

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

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

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

Answer:

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

Example 7

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

Solution: Let's make a drawing:

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

For the purpose of finding the volume of the body of revolution, it is enough to use the right half of the figure, which I shaded in blue. Both functions are even, their graphs are symmetrical about the axis, and our figure is also symmetrical. Thus, the shaded right part, rotating around the axis, will certainly coincide with the left unhatched part.

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

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

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

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

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:

Example 2

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

This is a do-it-yourself example. Full solution and answer at the end of the lesson.

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

Example 3

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

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

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

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

First, let's look at the figure that is circled in red. When it rotates around the axis, a truncated cone is obtained. Denote the volume of this truncated cone by.

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

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

We use the standard formula for finding the volume of a body of revolution:

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

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

3) The volume of the desired body of revolution:

Answer:

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

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

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

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

Example 4

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

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

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 can also intersect the figure if it is the axis of symmetry of the figure.

Theorem 2.
, axis
and straight line segments
and

rotates around an axis
. Then the volume of the resulting body of revolution can be calculated by the formula

(2)

Proof. For such a body, the section with the abscissa is a circle of radius
, means
and formula (1) gives the desired result.

If the figure is limited by the graphs of two continuous functions
and
, and line segments
and
, moreover
and
, then when rotating around the abscissa axis, we get a body whose volume

Example 3 Calculate the volume of a torus obtained by rotating a circle bounded by a circle

around the x-axis.

R solution. The specified circle is bounded from below by the graph of the function
, and above -
. The difference of the squares of these functions:

Desired 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 , revolves around the base. Calculate the volume of the resulting body ("lemon" by Cavalieri).

R solution. Place the parabola as shown in the figure. Then its equation
, and
. Let's find the value of the parameter :
. So, the desired volume:

Theorem 3. Let a curvilinear trapezoid bounded by the graph of a continuous non-negative function
, axis
and straight line segments
and
, moreover
, rotates around an axis
. Then the volume of the resulting body of revolution can be found by the formula

(3)

proof idea. Splitting the segment
dots

, into parts and draw straight lines
. The whole trapezoid will decompose into strips, which can be considered approximately rectangles with a base
and height
.

The cylinder resulting from the rotation of such a rectangle is cut along the generatrix and unfolded. We get an “almost” parallelepiped with dimensions:
,
and
. Its volume
. So, for the volume of a body of revolution we will have an approximate equality

To obtain exact equality, we must pass 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.

Remark 1. In Theorems 2 and 3, the condition
can be omitted: formula (2) is generally insensitive to the sign
, and in formula (3) it suffices
replaced by
.

Example 5 Parabolic segment (base
, height ) revolves around the height. Find the volume of the resulting body.

Solution. Arrange the parabola as shown in the figure. And although the axis of rotation crosses the figure, it - the axis - is the axis of symmetry. Therefore, only the right half of the segment should be considered. Parabola equation
, and
, means
. We have for volume:

Remark 2. If the curvilinear boundary of a curvilinear trapezoid is given by the parametric equations
,
,
and
,
then formulas (2) and (3) can be used with the replacement on the
and
on the
when it changes t from
before .

Example 6 The figure is bounded by the first arc of the cycloid
,
,
, and the abscissa axis. Find the volume of the body obtained by rotating this figure around: 1) the axis
; 2) axles
.

Solution. 1) General formula
In our case:

2) General formula
For our figure:

We encourage students to do all the calculations themselves.

Remark 3. Let a curvilinear sector bounded by a continuous line
and rays
,

, rotates around the polar axis. The volume of the resulting body can be calculated by the formula.

Example 7 Part of a figure bounded by a cardioid
, lying outside the circle
, rotates around the polar axis. Find the volume of the resulting body.

Solution. Both lines, and hence the figure they limit, are symmetrical about the polar axis. Therefore, it is necessary to consider only the part for which
. The curves intersect at
and

at
. Further, the figure can be considered as the difference of two sectors, and hence the volume can be calculated as the difference of two integrals. We have:

Tasks for an independent solution.

1. A circular segment whose base
, height , revolves 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 x-axis. Find the volume of the body, which is obtained in this case.

4. Figure bounded by lines
and
rotates around the x-axis. Find the volume of the body of revolution.

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 \u200b\u200bthe figure, the volume of the body of rotation, the length of the arc, the surface area of rotation, and much more. So it will be fun, please be optimistic!

- around the x-axis;
- around the y-axis.

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

flat figure around an axis

Calculate the volume of the body obtained by rotating the figure bounded by lines around the axis.

Solution: As in the area problem, the solution starts with a drawing of a flat figure. That is, on the plane it is necessary to build a figure bounded by lines , , while not forgetting that the equation defines the axis . How to make a drawing more rationally and faster can be found on the pages Graphs and Properties of Elementary Functions and . This is a Chinese reminder and I don't stop at this point.

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:

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

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

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

Calculate the volume of the body of revolution using this formula: