Rotation around the AU axis. How to calculate the scope of rotation using a specific integral? Calculation of body volume formed by rotation of a flat shape around the axis

C is contained in the interval. Thus, we again got a Langundree form of an additional member. 5. Conclusion. In the course work, the definitions of a certain and incompatible integral and its types are given, issues of some application of a particular integral are considered. In particular, the Valles formula, which has historical importance, as the first representation of the number P in the form of a limit easily calculated ...

the radiated integral of the type of function is numerically represents the area of \u200b\u200bcurvilinear trapezion of bounded curves x \u003d 0, y \u003d a, y \u003d b and y \u003d (Fig. 1). There are two methods for calculating this area or a specific integral - method of trapezoids (Fig. 2) and the middle rectangle method (Fig. 3). Fig. 1. A curvilinear trapezium. Fig. 2. Method of trapezium. Fig. 3. Method of middle rectangles. According to methods ...

N (increasing the number of integrations) increases the accuracy of the approximate calculation of the integrals task to laboratory work 1) Write programs for calculating a specific integral methods: medium, right rectangles, trapezoids and the Simpson method. Perform the integration of the following functions: 1. f (x) \u003d xf (x) \u003d x2 f (x) \u003d x3 f (x) \u003d x4 on the segment with a pitch, 2. f (x) \u003d f (x) \u003d f (x ) \u003d ...

... (TABL procedure) and integral. 4. Conclusion and conclusions. Thus, it is obvious that when calculating certain integrals with the help of quadrature formulas, and in particular, the Chebyshev formula does not give us exact value, but only approximate. In order to close as much as possible to a reliable value of the integral, you need to be able to choose the right method and the formula that will be calculated. Same...

flat shape around the axis

Example 3.

Dana flat figure limited by lines ,,

1) Find the area of \u200b\u200ba flat figure limited by these lines.

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

Attention! Even if you want to get acquainted only with the second item, first before Read the first!

Decision: The task consists of two parts. Let's start with the square.

1) Perform drawing:

It is easy to see that the function sets the upper branch of the parabola, and the function is the lower branch of the parabola. Before us is a trivial parabola that "lies on the side."

The desired figure, the area of \u200b\u200bwhich is to be found, shaded in blue.

How to find the area of \u200b\u200bthe figure? It can be found "ordinary" way. Moreover, the area of \u200b\u200bthe figure is like the amount of the area:

- on the segment;

- On the segment.

Therefore:

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

How to go to reverse functions? Roughly speaking, you need to express "X" through "Irek". First we will deal with parabola:

This is enough, but make sure that the same function can be removed from the bottom branch:

With straight, everything is easier:

Now we look at the axis: Please, periodically tilt your head to the right of 90 degrees along the course of the explanation (this is not a joke!). The figure we need lies on the segment, which is marked with a red dotted line. At the same time, the straight line is located above the parabola, and therefore the area of \u200b\u200bthe figure should be found on the formula already familiar to you :. What changed in the formula? Only the letter, and nothing more.

! Note : Asian integration limits It should be arrangedstrictly bottom up !

Find area:

On the segment, so:

Please note how I implemented integration is the most rational way, and in the next point of the task will be clear - why.

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

The initial integrand is obtained, it means that the integration is made correctly.

Answer:

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

Redrawing drawing a little in another design:

So, the figure, shaded in blue, rotates around the axis. As a result, it turns out a "hung butterfly" that spins around its axis.

To find the volume of the body of rotation, we will integrate along the axis. First you need to go to reverse functions. This is already done and described in detail in the previous paragraph.

Now we til off the right again and we study our figure. Obviously, the volume of the body of rotation should be found as a difference in volumes.

Rotate the figure, circled in red, around the axis, resulting in a truncated cone. Denote this volume through.

Rotate the figure, circled with green, around the axis and denote through the volume of the body of rotation.

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

We use the formula for finding the volume of the body of rotation:

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

But the advantage of integration, which I recently spoke is much easier to find than to pre-build a reintroduct function in the 4th degree.

Answer:

Note that if the same flat figure rotate around the axis, then it will turn out a completely different body of rotation, of the other, naturally, the volume.

Example 7.

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

Decision: Perform drawing:

Along the way I get acquainted with the graphs of some other functions. Such an interesting schedule of an integer function ....

For the purpose of finding the volume of the body of rotation, it is enough to use the right half of the figure, which I shared in blue. Both functions are even, their graphs are symmetrical about the axis, symmetrical and our figure. Thus, the shaded right part, rotating around the axis, will certainly coincide with the left-handed part. or . In fact, I myself am always insured, substituting a couple of points of schedule in the found function found.

Now we creep your head right and notice the next thing:

- on the segment over the axis there is a graph of a function;

It is logical to assume that the volume of the body of rotation needs to be searched as the amount of the volume of the bodies of rotations!

We use the formula:

In this case.

The volume of the rotation body can be calculated by the formula:

In the formula, the integral is necessarily present. It was so necessary - everything that spins in life is associated with this constant.

How to arrange the limits of integration "A" and "BE", I think it is easy to guess from the drawing made.

Function ... What is this function? Let's look at the drawing. Flat figure is limited to the Parabolys top schedule. This is the function that is meant in the formula.

In practical tasks, a flat figure can sometimes be located below the axis. This does not change anything - the integrated function in the formula is built into the square: so the integral is always nonnegatory that is very logical.

Calculate the scope of rotation using this formula:

As I have already noted, the integral is almost always simple, the main thing is to be attentive.

Answer:

In response, you must define the dimension - cubic units. That is, in our body of rotation approximately 3.35 "cubes". Why it is cubic units? Because the most universal wording. Cubic centimeters can be cubic meters, there may be cubic kilometers, etc., this is how many green men your imagination will be placed in a flying plate.

Example 2.

Find the volume of the body formed by the rotation around the shape axis limited by lines ,,

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

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

Example 3.

Calculate the volume of the body obtained when rotating around the abscissa axis of the figure limited lines ,, and

Decision: Show a flat figure in the drawing, limited by lines ,,,, not forgetting that equation is the axis:

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

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

First consider the figure, which is circled in red. With its rotation around the axis, a truncated cone is obtained. Denote the volume of this truncated cone through.

Consider a figure that is circled with green. If you rotate this figure around the axis, you will also get a truncated cone, only a little smaller. Denote its volume through.

And, obviously, the difference in volumes is exactly the volume of our "bagel".

We use the standard formula for finding the volume of the body of rotation:

1) The figure circled in red is limited from above straight, so:

2) The figure visited green is limited from above straight, so:

3) the volume of the original body of rotation:

Answer:

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

The decision itself is more often arranged in short, approximately in such a spirit:

Now a little rest, and tell about the geometric illusions.

People often have illusions associated with the volume, which was noticed by Perelman (another) in the book Entertaining geometry. Look at the flat figure in the tried task - it seems to be small in the area, and the volume of the body of rotation is slightly over 50 cubic units, which seems too much. By the way, the average person in his entire life drinks a liquid with a room with an area of \u200b\u200b18 square meters, which, on the contrary, seems too small.

In general, the education system in the USSR was really the best. The same Book of Perelman, published in 1950, develops very well, as a humorist said, consolidate and teaches to look for original non-standard solutions to problems. Recently, some chapters read with great interest, recommend, accessible even for humanitarian. No, you do not need to smile that I offered an impact pastime, erudition and a wide range of communication - a great thing.

After lyric retreat, it is relevant to solve the creative task:

Example 4.

Calculate the volume of the body formed by the rotation relative to the axis of the flat figure limited by the lines, where.

This is an example for an independent solution. Please note that all matters occur in the strip, in other words, the finished integration limits are actually given. Properly draw graphs of trigonometric functions, remind the material of the lesson about geometric chart transformations : If the argument is divided into two: then the graphs are stretched by ledging twice. It is advisable to find at least 3-4 points according to trigonometric tables In order to more accurately perform the drawing. Complete solution and answer at the end of the lesson. By the way, the task can be solved rationally and is not very rational.

Use of integrals to find the volume of bodies of rotation

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

specific mathematical knowledge is hampered by the understanding of the principles of the device and the use of modern technology. Each person in his life has to perform quite complex calculations, to use common technique, to find in reference books to apply the necessary formulas, to make simple algorithms for solving problems. In modern society, more and more specialties requiring a high level of education are associated with the direct use of mathematics. Thus, for a schoolboy, mathematics becomes a professional meaningful subject. The leading role belongs to mathematics in the formation of algorithmic thinking, brings up the ability to act according to a given algorithm and design new algorithms.

Studying the topic of application of the integral to calculate the volume of bodies of rotation, I offer students at optional classes to consider the topic: "The volumes of rotation bodies with the use of integrals." Below, we give methodical recommendations for consideration of this topic:

1. Flat shape location.

From the course of algebra, we know that the concept of a certain integral has led to a practical task ... "Width \u003d" 88 "Height \u003d" 51 "\u003e. Jpg" width \u003d "526" height \u003d "262 src \u003d"\u003e

https://pandia.ru/text/77/502/images/image006_95.gif "width \u003d" 127 "height \u003d" 25 src \u003d "\u003e.

To find the volume of the body of rotation formed by the rotation of the curvilinear trapezium around the OX axis bounded by the interrupt line y \u003d f (x), the OX axis, direct x \u003d a and x \u003d b calculated by the formula

https://pandia.ru/text/77/502/images/image008_26.jpg "width \u003d" 352 "height \u003d" 283 src \u003d "\u003e y

3. Cylinder volume.

https://pandia.ru/text/77/502/images/image011_58.gif "width \u003d" 85 "height \u003d" 51 "\u003e .. gif" width \u003d "13" height \u003d "25"\u003e .. jpg " width \u003d "401" Height \u003d "355"\u003e Cone is obtained by rotating the rectangular triangle ABC (C \u003d 90) around the OX axis on which the speaker is lying.

Cut AV lies on a straight line y \u003d kx + C, where https://pandia.ru/text/77/502/images/image019_33.gif "width \u003d" 59 "height \u003d" 41 src \u003d "\u003e.

Let a \u003d 0, b \u003d h (the height of the cone), then vhttps: //pandia.ru/text/77/502/images/image021_27.gif "width \u003d" 13 "height \u003d" 23 src \u003d "\u003e.

5. Punction of a truncated cone.

The truncated cone can be obtained by rotating the rectangular trapezium of the AVD (CDOX) around the OX axis.

Cut AB lies on a straight line y \u003d kx + C, where , C \u003d R.

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

Thus, the line has the appearance of https://pandia.ru/text/77/502/images/image027_17.gif "width \u003d" 303 "height \u003d" 291 src \u003d "\u003e

Let a \u003d 0, b \u003d h (n- height of a truncated cone), then https://pandia.ru/text/77/502/images/image030_16.gif "width \u003d" 36 "height \u003d" 17 src \u003d "\u003e \u003d. .

6. Bowl.

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

https://pandia.ru/text/77/502/images/image034_13.gif "width \u003d" 13 "height \u003d" 16 src \u003d "\u003e x r.

Definition 3. The body of rotation is the body obtained by rotating a flat figure around the axis that does not cross the figure and lying with it in the same plane.

The axis of rotation can and cross the figure, if it is the axis of the symmetry of the figure.

Theorem 2.
, axis
and straight cuts
and

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

(2)

Evidence. For such a body, the cross section with the abscissa - this is a circle of radius
So
and formula (1) gives the required result.

If the figure is limited to the graphs of two continuous functions
and
, and straight cuts
and
Moreover
and
, when rotating around the abscissa axis, we obtain the body, the volume of which

Example 3. Calculate the volume of the torus obtained by the rotation of the circle limited by the circle

around the abscissa axis.

R measure. The specified circle bottom is limited by a graph
, and from above -
. The difference of squares of these functions:

The desired volume

(A graph of the integrand is the upper part-friendly, so the integral written above is the semicircular area).

Example 4. Parabolic segment
, and high , rotates around the base. Calculate the volume of the resulting body ("Lemon" Cavalieri).

R measure. Parabola is placed as shown in the figure. Then its equation
, and
. Find the value of the parameter :
. So, the desired volume:

Theorem 3. Let a curvilinear trapeze, limited by a chart of a continuous non-negative function
, axis
and straight cuts
and
Moreover
rotates around the axis
. Then the volume of the receiving body of rotation can be found by the formula

(3)

The idea of \u200b\u200bproof. Small cut
points

, part and spend direct
. The whole trapeze will decompose on strips, which can be considered approximately rectangles with the base.
and height
.

The cylinder is obtained when rotating such a rectangle, we will cut through the forming and unfold. We get "almost" parallelepiped with dimensions:
,
and
. Its volume
. So, for the volume of the body of rotation, we will have approximate equality

To obtain accurate equality, you need to go to the limit when
. The amount written above is the integral amount for the function
Therefore, in the limit, we obtain the integral from formula (3). Theorem is proved.

Note 1. In Theorems 2 and 3 Condition
you can omit: Formula (2) is generally insensitive to the sign
, and in formula (3) enough
replaced by
.

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

Decision. Place a parabola as shown in the figure. And although the axis of rotation crosses the figure, it is the axis - is the axis of symmetry. Therefore, it is necessary to consider only the right half of the segment. Parabolla equation
, and
So
. We have for volume:

Note 2. If the curvilinear boundary of the curvilinear trapezium is set by parametric equations
,
,
and
,
you can use formulas (2) and (3) with replacement on the
and
on the
when it changes t. from
before .

Example 6. Figure is limited to the first arch cycloids
,
,
, and the abscissa axis. Find the volume of the body obtained by the rotation of this figure around: 1) the axis
; 2) axis
.

Decision. 1) general formula
In our case:

2) general formula
For our figure:

We offer students to independently carry out all the calculations.

Note 3. Let the curvilinear sector limited to the neur-rive
and rays
,

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

Example 7. Part of the shape limited cardioid
circumference
rotates around the polar axis. Find the volume of the body that it turns out.

Decision. Both lines, and therefore the figure that they limit is symmetrical with respect to the polar axis. Therefore, it is necessary to consider only the part for which
. Curves intersect
and

for
. Further, the figure can be considered as a difference of two sectors, which means the volume to calculate as the difference between the two integrals. We have:

Tasks For an independent solution.

1. Circular segment whose base
, height , rotates around the base. Find the scope of rotation.

2. Find the volume of the paraboloid of rotation, the base of which and the height is equal .

3. Figure limited by Astroide
,
rotates-smiling around the abscissa axis. Find the volume of the body that is obtained.

4. Figure limited lines
and
rates around the axis of the abscissa. Find the scope of rotation.