3 Supplements of subintegrals

3.1 Theoretical introduction

The addendums of the sub-base integral to the top are low geometric buildings that department of mechanics.

3.1.1 Calculating the area of ​​the flat plate

Visible thin material plate D, roasted in the area Ooh. Area S The whole plate can be found behind the additional subintegral for the formula:

3.1.2 Static moments. Center of the flat plate

A static moment M x schodo osi Ox material point P(x;y) to lie near the area Oxy i maє masu m, To be called dobutok masi point on the ordinate, tobto. M x = my... Similarly, there is a static moment M y schodo osi Ouch: ­ ­ ­ M y = mx. Static moments flat plate with surface fissure γ = γ (x, y) are calculated according to the formulas:

Yak vidomo z mechanics, coordinates x c , y c the center of the mass of flat material systems and start with the following ambitions:

de m- Masa system, and M xі M y- Static moments of the system. Flat plate masa m Beginning by formula (1), the static moments of a flat plate can be calculated according to formulas (3) and (4). Todi, according to formulas (5), we can deduce the viraz for the coordinates to the center of the mass of the flat plate:

Typical rozrahunok to take revenge on two tasks. The skin department has a flat plate D, surrounded by lines, indicated for the mind. G(x, y) - surface area of ​​the plate D... For the whole plate to know: 1. S- Area; 2. m- Masu; 3. M y , M x- Static moments of the axes Oyі Oh for certain; 4., - Coordinate to the center of the mass.

3.3 Procedure for submitting a sample size

When you see a skin condition, it is necessary to: 1. Make a chair for a given area. Vibrate the coordinate system, where the subintegrals are calculated. 2. Write down the area in the view of the system of irregularities at the vibration of the coordinate system. 3. Count the area S ta masu m plates for formulas (1) and (2). 4. Calculate static moments M y , M x for formulas (3) and (4). 5. Calculate the coordinates to the center of the mass according to the formulas (6). Apply the center of the mass to the chair. We have a visual (yakisny) control of the results. Numbers from may be recognized as three digits.

3.4 Attach a sample file

Zavdannya 1. Plate D surrounded by lines: y = 4 – x 2 ; NS = 0; y = 0 (x ≥ 0; y≥ 0) Surface area γ 0 = 3. Decision. The region specified in the problem is surrounded by a parabola y = 4 – x 2, the axes of coordinates і lie at the first quarter (Fig. 1). Zavdannya virishuvatimemo in Cartesian coordinate systems. The qia area can be described by a system of irregularities:

Small. 1

Area S plates for doors (1): So yak a plate is single-sided, її masa m = γ 0 S= 3 · = 16. Behind the formulas (3), (4) we know the static moment of the plate: The coordinates of the center mass are found for the formula (6): View: S ≈ 5,33; m = 16; M x = 25,6; M y = 12; = 0,75; = 1,6.

Zavdannya 2. Plate D surrounded by lines: NS 2 + at 2 = 4; NS = 0, at = NS (NS ≥ 0, at≥ 0). Superficiality γ (x, y) = at. Decision. The plate is lined with a stake and straight lines passing through the cob of coordinates (Fig. 2). To this end, for the purpose of resolving tasks, the polar coordinate system is manually victorious. Polar kut φ change from π / 4 to π / 2. Promin, conducting from the pole through the plate, "enter" to it at ρ = 0 and "enter" on the stake, equal to: NS 2 + at 2 = 4 <=>ρ = 2.

Small. 2

Otzhe, the given area can be recorded by the system of irregularities: The area of ​​the plate is known by the formula (1): We know the masu of the plate for the formula (2), having submitted γ (x, y) = y = ρ sin φ :
For calculating the static moments of the plate, the Vikorist formula (3) and (4):
The coordinates of the center of the mass are visible according to the formulas (6): View: S ≈ 1,57; m ≈ 1,886; M x = 2,57; M y = 1; = 0,53; = 1,36.

3.5 Registration of the call

At the beginning of the day, there are all the vicinities of the rosary, neatly the vicinities of the armchair. Numbers from may be recognized as three digits.

Guidedly, the butt is assigned to the center of the power by the method of podil yogo on the okremi tila, the center of the masses.

Butt 1... Viznachiti coordinates to the center of the mass of the one-sided plate (Fig. 9). The sizes are set at the millimeters of the baby 9.

Decision: The axis of coordinates і is shown. The plate can be broken into pieces, installed with three rectangulars. For the cutaneous rectum, a diagonal is carried out, the points of overturning are the beginning of the position of the centers of the mass of the cutaneous rectus. Accepted coordinate systems are awkward due to the value of the coordinates of the number of points. And itself:

(-1; 1), (1; 5), (5; 9). The area of ​​the skin is as follows:

; ; .

The area of ​​the entire door plate:

For the assignment of coordinates to the center of the mass of a given plate, it is stagnant virazi (21). Apparently, the value of all types of values ​​for a given relative, is recognized

Depending on the value of the coordinates to the center of the mass of the plate, it is also a point on the little one. Yak can be seen, the center of the masi (geometric point) of the plate is located in the posture of її between.

How to add... Tsei sposіb є in a frequent way in a way. Win can be stashed to til, as may be virizi (empty). Moreover, without the virіzanoi part of the position to the center of the mass tila vіdomo. It is understandable, for example, for such a method.

stock 2. The position of the center of the mass of the vagi of the round plate by the radius R, de viriz by the radius r (Fig. 10). Get there.

Decision: Yak bachimo, from fig. 10 the center of the mass of the plate should lie on the axis of the symmetry of the plate, so that on a straight line, the center of the mass of the plate lies on the axis of the symmetry of the plate. In such a rank, in order to value the position of the center of the mass of the central plate, it is necessary to assign only one coordinate, and some other coordinate will be rotated on the axis of symmetry and temporarily zeros. Shown are the coordinate axes,. Acceptable, the plate is stored in two til - a basic stake (nachebto without viriza) and tila, like nibi viconane with virizom. In the adopted coordinate systems, the coordinates for the values ​​of the door are:. ... The back area of ​​the entire building is diversified between the areas of the first and the other building, but itself

Calculate the values m, and it is necessary to use formulas (4), (5) and (7). As a result, we will formulas for coordinates to the center of a thin plate :

Butt 4 (calculating coordinates to the center of the one-sided plate)

Know the coordinates of the center of one-sided figurines, surrounded by lines.

Having stayed with a figuru, of course, it’s geometrically won є symmetrically straight. right-handed Todi for vidomimi physical authorities the center is masculine, which is on the axis of symmetry,

Calculate the warehouse static moment and the vicious formulas (4) and (5):

;

Suggestion: C.

Supplements for third-party integrals

Programs for third-party integrals are analogous to supplements of sub-integrals, ale only for trivial ones.

If you are victorious, one of the powers of the consumer integral (about this meaning from the function, which can also be used as one), then go formula for calculating the volume of any spacious building :

Write down the formula for the exchange through a third-party integral and a calculable third-party integral cylindrical coordinates:

Vidpovid: (one obsyagu).

Formula for calculating the mass of a trivial object, scho loan volume V, maє viglyad:

(13)

Here the whole thing is about the power of the oil.

Butt 6 (calculating the weight of the trivial til)

Know Masu Kuli Radiusu R as the proportions of the cube from the city to the center and on the same city from the road k.

V: elementary volume that.

Amazingly, when the three-time integral was calculated, there were some internal integrals, some of the internal integrals were found to be fallow from the winter integrals.

View: (one masi).

Mechanical characteristics for obsjagu V(Static moments, moments of energy, coordinates to the center of the mass) are calculated according to the formulas, such as

stocked for analogous formulas for double-sided tl.

Elementary static moments and moments of energy from the coordinate axes:

elementary moments of energy coordinate areas that point is a cob of coordinates:

Dal, just count the mechanical characteristics of all obsyagu V, It is necessary to provide the elementary completions of the general characteristics for all parts of the rozbittya (the characteristics of the power of additivity have been calculated), but we can go to the middle in the sum, but went for the mind, but it is necessary to change the points (tightening all the time) The process is described as the integration of the elementary addition of the mechanical characteristics, which can be calculated, for a long time V.

As a result, come the offensive formulas for calculating static moments M and moments of energy I trivial :

As a matter of fact, the formulas yak vicoristovuvati yak are ready, and yak vivoditi in virіshuvanіy tasks.

Apply 7 (calculating the mechanical characteristics of trivial til)

Know the moment of inertia of a single-sided cylinder, the height of h that radius basis R, as well as the axis, should be set up with the diameter of the base.

We know d for a certain point of the cylinder:

view from the point with coordinates to the axis – perpendicular line drawn from the point to the axis . Let the area be perpendicular to the axis so that the point is to trace the area. If it’s straight, it’s overflowing, and if it’s right, it’s perpendicular . The dawn, straight, will be perpendicular to the axis, and will appear between the dots and will be d... It is calculated by the given formula in two dots.

– ascribing to the center of gravity of a flat intermeshed figuri... Bagato reading intuitive intelligence, which is also the center of gravity, protest I recommend repeating the material from one lesson analytical geometry de i rozibrav zavdannya about the center of gravity and available forms physical sense the whole term.

Independent and control systems for verification, as a rule, will display the simplest type - the flat is surrounded odnoridna figur, to post a post-mortem physical scholarship - a sklyan, a tree, a tin chavunna igrashka, a heavy dignity is too much. Dal for talking about mova pide tilki about such figuri =)

The first rule is that the simplest butt: how flat figure maє center of symmetry, then it is the center of gravity of the figuri community... For example, the center of a round one-sided plate. Logically і life іѕ zoosuіlo - the mass of such figuri "is justly distributed on all sides" to the center. Verti - I don't want to.

However, in suvorikh realities, you are unlikely to get licorice delicious chocolate That will have a chance to get used to a serious kitchen tool:

Coordinate the center of gravity of a one-sided flat interlaced figure to be insured according to such formulas:

, abo:

, de - Area of ​​the region (figuri); for a short call:

, de

The integral is cleverly called "iksovim" integral, and the integral is called "іgrekovim" integral.

Note-follow-up : for flat bounded heterogeneous figuri, the degree of which is given by the function, folding formulas:
, de - Masa figuri;at once one-sided stench to say goodbye to the guiding formulas.

On the formulas, vasne, all the novelty and end, the solution is your price virishuvati sub-integrals, before the speech, at once it is a miraculous opportunity to try to get to grips with your technique. A thoroughness, yak vidomo, dumb mezhi =)

Letting go with a heavy-duty portrayal of parabolas:

Butt 1

Know the coordinates of the center of the wagi of one-sided flat figurines, surrounded by lines.

Decision: line here elements: set the abscis, and the ryvnyannya - a parabola, it’s easy that I’ll quickly go for help geometric transformation of graphs:

– parabola, Shunted 2 units left and 1 unit down.

I show the whole chair at once with a ready-made point to the center of the vagi figuri:

Rule to a friend: yakscho u figuri isnu all simetry, then the center of gravity of the given figuri must lie on the axis.

Our vipadku figure is symmetrical. straight That is, in fact, the same is known to the "iksovu" coordinate of the point "eat".

Likewise, beastly respect, as vertically the center of heavy measures is closer to the abscissa axis, there is a picture of a mass there.

So, maybe, not everything until the end of the zoos, but also the center of the vagi: be a weasel, put your finger on the top and put a speck on the shaded "foot". In theory, fasting is innocent.

We know how to coordinate the center of gravity of the figuri for the formulas de.

The procedure for bypassing the region (figuri) is obvious here:

Uwaga! Visnachamosya with the best possible traversal order once- I vikoristymo yogo for all integraraliv!

1) The spatka is numbered over the area of ​​the figuri. Through the extraordinary simplicity of the integration, the solution can be arranged compactly, smut, not get lost in the calculation:

Marveling at the armchair and prikidayamo on the clitin area. Have a go!

2) The X coordinate to the center of gravity is already known by the "graphical method", so you can pay for symmetry and go to the offensive point. However, it’s not so robust, after all, it’s great value, but the decision to reject from the formula “pick up the formula”.


Believe me, here you can get by with the vinyatkovy numbered - one does not necessarily bring the fractions to the spilny denominator of the calculator.

In this rank:
, Scho and it is necessary to correct.

3) We know the ordinate of the center of gravity. Numerous "іgroviy" іntegral:

And the axis here would be hard without a calculator. I will comment on all kinds of issues, as a result of a lot of bugs to go to 9 members, and the deyak of them are a little more. More donations I vaccinated asleep (yak tse zzvychay it is accepted by robiti in some vipadkah) and immediately writing down the pidsumkovo sum.

As a result:
it is even more like the truth.

On the final stage, there is a speck on the armchair. For a wash, you don't need any armchair, but in a big way, I want to imagine a figuru. Natomist є insane plus - visual and effective adjustment to the result.

View:

Step two butt of an independent solution.

Butt 2

Know the coordinates of the center of the wagi of one-sided flat figurines, surrounded by lines

Before the speech, if you notice that the parabola has been removed and the dots have been shaken, then here and indeed you can do without a chair.

І foldable:

Butt 3

Know the center of the wagy of one-sided flat figuri, surrounded by lines

At times of difficulties due to prompting graphs to vivchit (repeat) lesson about parabolas that / abo Appendix No. 11 of the statti Connectors for teapots.

Educational insight into the lesson for a lesson.

In addition, a dozen or more similar butts can be found in the general archives on the side Preparing a solution in math.

Well, I can’t help but please the lovers of all kinds of mathematics, as they often ask me for a choice and important work:

Butt 4

To know the center of the wagy of one-sided flat figurines, surrounded by lines. Figuru and the center of gravity of the image on the armchair.

Decision: mind of a set of tasks already categorically vimagaє vikonannya armchair Aje vimoga not nastilka і formally! - It’s great to see people in the mind because of the average level of training:

Directly rozsіkє colo into 2 parts, (Div. line of irregularities) vkazє on those who are about the little shading little piece.

The figure is symmetrical and obviously straight (depicted by a dotted line), the center of the heavy is to blame for lying on the line. It is obvious that the coordinates of the behind the module... Vidminny orintir, as a practical matter to see pomilkov!

Now nasty novina =) On the horizon loom a little-accepted integral from the root, which was taken from Prikladi №4 to the lesson Effective methods of integration of integration... I know how it will get dirty there. Building used, through explicitness cola Obviously, prote is not so simple. Rivnyannya straightforwardly transformed into a viglyad that integration tezh viydut not tsukrovi (hocha fanati trigonometric integrals evaluate). The link with the cymbals usually zupinititsya on the Cartesian coordinates.

The order of bypassing figuri:

1) The area of ​​the figuri is calculated:

First Integral Rationally Taking sign for differential:

And in another integration, a standard replacement is carried out:


Numerous new integer integration:

2) We know.

Here at the 2nd Integral I know buv vikoristaniy method of bringing the function to the differential sign... Make the decision to restore the optimal (on my thought) accept the development of typical integrals.

The song is difficult and trivial; (pam'yataєmo, scho points I don’t know! ) і іn acceptably more morally satisfied with the known meaning.

3) Vyhodyachi from the analysis carried out earlier, overcrowded, scho.

Vidminno:

Imaginable point on the armchair. According to the formulation, wash and write down її yak residual vidpovid:

More information for independent version:

Butt 5

To know the center of the wagy of one-sided flat figurines, surrounded by lines. Viconati armchair.

At the same time, it’s interesting that the figure is set to finish the little ones, and if there’s some pardon, then if you’re going to “don’t eat” in the region. Scho, bezperechno, good from the point of view of control of the decision.

Exceptional design for a lesson.

Inodi buvak docilnym transition to polar coordinates at subintegrals... Tse vid figuri. Shukav-shukav with a far-off butt, ale I don't know, so I will demonstrate the solution to the 1st demo-task of the given lesson:


I guess, well, in that case, we went to polar coordinates, took the order to bypass the region і virahuvali її square

Let's know the center of gravity of the figuri community. The scheme is the same: ... The meaning can be seen directly from the armchair, and the "iksova" coordinate is changed to the nearest axis of ordinates;

In the integrals, the vicorist standard formulas for the transition are:


Ymovirno, shvidshe for everything, they did not have mercy.