Nehay was given material tilo, but a spacious area P, memorized by Masoy. It is necessary to know the weight of the whole body for the reason, which in the skin point R € P in the house of the distribution of the weight. Rozib'єmo area P on cubes, but not over-crooked (tobto mayut volume) parts with liabilities are obvious. In the skin of the partial areas ft *, the vibration point is P *. Acceptable approximately, at the border of the private area ft * the power of the post and the road / * (P *). Todi mass of Atk of a part of the building will vary with the approximate equality of Atpk and the mass of all this will be approximately expensive. Cartesian coordinates Enumeration of the consumer integral in cylindrical and spherical coordinates Nekhai d - the largest of the diameters of the subdivisions of Yakshho at d - * 0 sum (1) there is no end of the border, as it is not possible to lay it down in the same way as the vibration of the area ft in the subtotals ft *, then the boundary will be taken for the mass m of the given floor, but in the closed cubed area ft, the function of Rozib'єmo ft is interconnected on n cubic parts, but it does not change, but the relations are meaningfully through the case. At the dermal part of the sub-area P *, with a high rank, the point Pk (xk, yk, zk) is vibrated and the warehouse integrated sum Nehay d is the largest diametr in the sub-area of the Viznachennya. If d About the integral sumi but may be between, as not to lay down in any way of developing the area L on the part of the pied-area P *, but not in the choice of points Pk € P *, then the boundary is called three-part integrals f (x) functions the area Q and is denoted by the symbol. Theorem 6. If the function f (x, y, z) is uninterrupted in a closed area, but as P is enclosed, then it is integrated in the whole area. The power of the third integrals The power of the third integrals is analogous to the powers of the subordinate integrals Pererakhumo main of them. Don't know the function to integrate into the area where you will be able to. L. 1. Linearity. At the same time, the function is called integrated in the Q. This rank, according to the designation, can be turned to tasks about calculating the mass of data, respect, that between (2) є required integral dyt function dylyanz (P) according to P. Element volume dv rectangular coordinates. de a і (3 - most of the speeches of the post-lynch. you are in the area P, then 3. If / (P) = 1 in the area P, then de V - the volume of the area Q. The function / (P) is uninterrupted in the closed cubed area ft і М і t - її best least significant y ft, de V - general area ft. 5. Additiveness. If the area ft is split into areas where it is cubed, without external internal points, f (P) is integrated in the area ft, then f (P) is integrated into the skin of the areas ft | ta ft2, with 6. Mean value theorem. Theorem 7 (about the mean). If the function f (P) is without interruption in a closed cuboidal space ft, then there is a thin Ps € ft, so, if the formula de V is valid - the total area ft (guess, that the area is unreliable). § 7. Calculation of the consumer integral in Cartesian coordinates Yak i pid hour of calculation subintegrals, On the right, it should be built up to the number of repeated integrals. Supposedly, the function is uninterrupted in the singing area ft. 1st vipadok. Area ft є by rectangular parallelepiped, which is designed for the area yOz at rectangular i2; Todi otrimaєmo Zamynyuchi subintegral through repeated, residually otrimaєmo In such a rank, if the region P is a rectangular parallelepiped, we started counting the third integral before the last counting of three integral integrals. Formula (2) can be rewritten in the view of the rectangle є is the orthogonal projection of the parallelepiped P onto the area xOy. 2nd vipadok. Now the region Q is clearly visible, so that it flanks surface 5, be it a straight line, parallel to the axis Oz, not more than at two points, or along the whole length (Fig. 22). Leave z = tpi (x, y) equal to the surface 5, which will enclose the area P below, and the surface S2, will enclose the area P from the top, ma equal z = y). Do not hurt the surface S and S2 to be projected on the same area of the area xOy. Significantly її through D, and the curve, which is surrounded by її through L. Reshta between 5 til Q lie on a cylindrical surface with a set, parallel axes Oz, і from the curve L at the rolі straight. Even after the analogy with formula (3), we can take the region D of the area xOy є by a curved trapezoid, surrounded by two curves, then the subintegral in formula (4) can be reduced to a second one, and otherwise the formula is reversible. Fig-23 Butt. Count the length of the tetrahedron, surrounded by areas The projection of the tetrahedron on the area xOy serves as a tricycle, the statements are straight so x change from 0 to 6, and with fixed x (0 ^ x ^ 6) y change from 0 to 3 - | (Fig. 23). If it is fixed and x, then the point can be shifted vertically from the area to the area between 0 to 6 - x - 2y. Behind the formula we will recognize §8. Enumeration of the consumer integral in cylindrical and spherical coordinates The nutrition of the substitute for the consumer in the consumer integral is shown in the same way as in the other sub-integral. Let the function / (z, y, z) be continuous in the closed area ft, but the functions are uninterrupted at once with their private private ones of the first order in the closed cubed area ft *. Supposedly, the function (1) establishes the unambiguous appearance between the points rj, () the area ft *, from one side, and all the points (z, y, z) of the area ft - from the other. That is the correct formula for replacing the changes in the consumer integration - de - Jacobian system of functions (1). As a matter of fact, when calculating third-party integrals, it is often used instead of rectangular coordinates by cylindrical and spherical coordinates. 8.1. The third integral in the cylindrical coordinates In the cylindrical coordinate systems of the position of the point P in the space, it is indicated by three numbers p, de p i (p is the polar coordinates of the projection P1 of the point P onto the area xOy, az is the direction of the point P (Fig. 24). scho in sistemі tsilіndrichnih coordinate koordinatnі poverhnі Potrіyny іntegral Vlastivostі potrіynih іntegralіv Obchislennya potrіynogo іntegrala in Cartesian coordinates Obchislennya potrіynogo іntegrala in tsilіndrichnih i sphericity coordinates vіdpovіdno opisuyut: circular tsilіndr, vіs yakogo zbіgaєtsya vіssyu of the Oz, napіvploschinu, i ploschinu, parallel to the area hou. The cylindrical coordinates are tied with the advancing Cartesian formulas (div. Fig. 24). For system (3), where the area ft is mapped to the area of mmo and formula (2) is the transition from the internal integral in rectangular coordinates to the integral in cylindrical coordinates. Tsei viraz of the element of volume can be trimmed from geometric mirkuvan. Rozib'єmo area P on the elementary piedoblasts with coordinate surfaces and the number of curved prisms is calculated (Fig. 25). At the same time, it is possible to take the volume in cylindrical coordinates for the element, the value will start. Application 1. To know the volume of the object, surrounded by surfaces. The surface of the surface is rewound along the line g, which is described by the system of ravines (cylindr), (area), Fig. 26 and її projection onto the area хОу by the system In this rank, Ob'єm, how to joke around, to be counted according to the formula (4), in which. The third integral at the spherical coordinates In the spherical coordinate systems, the position of the point P (x, y, z) in the open space is indicated by three numbers, de g - appear from the cob of coordinates to the point kut between the axis of the Oh and the projection of the radius vector of the point P on the plane and in - the cut between the axis Oz and the radius vector OR of the point P, which is drawn from the axis Oz (Fig. 27). Zrozumilo, scho. Coordinate surfaces at the ts_y coordinate systems: r = const - spheres with center on the cob of coordinates; ip = constnap_varea, go out from the axis Oz; в = const - circles & cones with weight Oz. Small. 27 It can be seen from the small that the spherical and Cartesian coordinates are tied by the advancing spins, the Jacobian functions are numbered (5). Otzhe, and formula (2) swells the viglyad Element volume in spherical coordinates - Viraz for the element volume can be trimmed from geometric worlds. An elementary area in the open space is clearly visible, surrounded by spheres of radii in r and d + dr, cones in i in + d $ and in areas. The area can be approached with a rectangular parallelepiped with vimir. ........ The calculation of the consumer integral in the Cartesian coordinates. The calculation of the consumer integral in the cylindrical and spherical coordinates. Application 2. Know the total number of the first conc. To the sphere Q From the third ryvnyannya known between change kuta 9: stars

Consumed Integrals. Calculated volume of tila.
The third integral at the cylindrical coordinates

Three days in the dean's office, lying down, at the trousers of Pythagoras,
In the hands of Fikhtengolts, there is a volume of trimmings,
Until nig they tied a third integral, and wrapped a corpse in a matrix,
And to replace the prayers of the hack by reading Bernoulli's theorem.

Consumed integrals - those that you can not be afraid of =) If you read the text, then you will be better off for everything. theory and practice of "zychanykh" integrarals and also subintegrals... And there, de secondary, not far and away:

First of all, what is there to be afraid of? Less integrated, more integrated.

Picked up from the record:

- the badge of an integral part;
- Pidintegralna function of three winters;
- Dobutok Differents.
- Integration area.

Especially for galuzi integration... Yaksho in subintegral won є flat figure, then here - spaciousness tilo, yak, yak vidomo, surrounded by bezlichchyu overhead... Such a rank, except for the guilt of guilt, orinntuvatisya in the main surfaces open to space and there are simple, simple armchairs.

Deyakі got confused, rozumіyu…. Unfortunately, the article cannot be called “useful integrals for teapots”, and it is often necessary for the nobility. Albeit nothing terrible - all the material of the wickedness is in the borderline accessible form and will be mastered from the nicest term!

What does it mean to count the third integral, and what does it mean?

Count the third integral - tse means know NUMBER:

At the simplest vipad, if, third-party integral, numerically,... First and foremost, according to zmіstom zmіstom integration, tvir one infinitely small volume of the elementary "tseglinka" tila. And the third integral yakraz i ob'єnuє all ci indefinitely small particles by region, inasmuch as who should be included in the integral (summarized) value of the volume of tila: .

Besides, the third-party integration is very important physical programs... Ale about the tse piznishe - at the 2nd part of the lesson, assignment the amount of sufficient third-party integration, for some functions, the view is constant and without interruption in the sphere. In tsіy statty it is possible to understand in detail the znakhozhennya obsyagu, as from my sub'active evaluation it is developed 6-7 times more often.

Yak virіshiti consumer integraral?

It appears logically from the previous point. It is necessary to visit order to bypass tila i go to repeat integrals... The message is sent lastly through three single integrals.

Yak bachite, the whole kitchen is just as bad as it gets subintegrals I see that at once we got a little extra space (roughly, apparently, hanging). I, melodiously, are rich enough for you already zdadavsya, how to see the use of integration.

Rosvієmo sumnіvi, scho lost:

Butt 1

Be a weasel, rewrite the stove to papir:

Give me feedback on the next meal. Do you know Vi, who are the surfaces to ask the price? Do you have an informal zmist tsikh ryvnyan? Why do you know Vi, how do you see the surface of the space in the open space?

If you are smart enough to say “shvidshe ni, nizh so”, then just explain the lesson in a straightforward way, you don’t get in!

Decision: vikoristova formula

For shhob z'yasuwati order to bypass tila i go to repeat integrals It is necessary (all genially simple) intelligence, very good. It’s great to take an armchair for such a reason in bagatioh vipadkas.

Behind the sinking tilo surrounded by a number of surfaces. Why do you want shikuvannya? I proponate the offensive order d_y:

Spatku conceivably parallel orthogonal projection of the floor to coordinate area... Having said the first time, what is the name of the projection, lol =)

If the project is carried out on a surfaces, which are parallel to the center of the axis. I guess, what a ryvnyannya of such surfaces do not take revenge on the letter "z"... The starter has three:

- Rivnyannya set the coordinate area, how to pass through the rope;
- Rivnyannya set the coordinate area, how to pass through the rope;
- івняння back flatness "Flat" straight parallel to the axis.

Shvidshe for everything, Shukana projection

You can, not all the residual sound, forgive me. See if you can go from the monitor screen and go directly to your transfer ( tobto. go in, see how you marvel at the 3-rd armchair from above)... Until the end of the day, it was more spacious to be in the endless trigonal "corridor" and projection onto the area of the newest, shaded by a tricycle.

I wrap up with special respect no more priuschennya about the projection and caution "nayshvidshe", "nyimovirnishe" bully vipadkovi. On the right, in the fact that not all surfaces have been analyzed yet, and it can be so, as if from them "see" a part of the tricycle. Yak with a stock butt ask sphere centered on a cob of coordinates with a radius smaller than one, for example, a sphere - її projection on the area (number ) I will not add the "hatched" shaded area, and the projection will not be called a tricycle (colo "zrіzhe" yomu gostrі kuti).

On the other side of the stage, there is a large, spacious armchair. We turned to the mind and wondered how the surfaces were overshadowed. Rivnyannya sets the coordinate area itself, and rivnyannya - parabolic cylinder, sewing above area and pass through the rope. In such a rank, the projection of the body is effective є by the tricycle.

Before the speech, here appeared overwhelm mind - in the new boulevard, it is not necessary to include a flat area, fragments of the surface, the abscissa axes sticking around, and so it slows down. Quite simply, in any case, we couldn’t have been able to projection at once - the tricycle "drawn" is deprived of going through the analysis of the rivnyannya.

A fragment of a parabolic cylinder is neatly depicted:

Pislya vikonannya armchair z bypass order any problems!

The order of bypassing the projection is important to the list. (with a lot of NAYZruchnisha, they are standing behind two-sided armchairs). Be afraid ABSOLUTELY LIKE W, yak i in subintegras! The laser pointer is that scanned flat area. Viberemo "traditional" 1st way to bypass:

Dal is taken into the hands of a charming lichtarik, marveling on a trivial armchair and strictly below the hill educate the patient. The promenade enter at the bottom through the area and go out through the surface. In this rank, the order of bypassing the body:

Let's move on to repeated integrals:

1) Take note of the "zeta" integration. Vikoristovuєmo Newton-Leibnitz formula:

Apparently the result of "Igor" Integral:

What happened? Along the way, the solution rang up to the subintegral, and itself - to the formula. volume of a cylindrical bar! Further good know:

I have a lot of respect for the rational technology of the third integration.

View:

The calculation can be recorded in one row:

In a different way, be safe - you will win the greediness at the bottom of the water, and if you have an important butt, there will be more chances to get pardoned.

Consider the important power supply:

Why do you need to rob an armchair, since the mind does not need a chair?

You can drink chotirma with paths:

1) Draw the same projection. The most popular option is that there is a possibility of two decent armchairs, do not go, rob the armchair offend. I recommend you go ahead.

2) Imagine a deprivation of tilo. It is good if the projection is awkward at the building. So, for example, a trivial armchair was removed from the selected butt. However, here і і minus - using 3D-pictures it is not manual to start the procedure for bypassing the projection, and in the best way I please b only people with good preparation.

3) Draw a deprivation of the projection. It’s not bad, but there is no obligation to add additional letters to comments, but the region is not surrounded by other sides. Unfortunately, the third option is often impertinent - if only there are great reasons for being tied with difficult difficulties. Put it like this on it is also clear.

4) Get around without a chair. In general, it is necessary to present only a little thought and comment on its form / form in writing. It’s important to go for a couple of simple people. Ale all the same, more beautiful than zrobiti would like a schematic little kid, oskіlki "goal" solution can be taken away.

Coming tilo for self-help:

Butt 2

For the help of the costly integral, count the volume of the body, surrounded by surfaces

In this context, the area of integration is set overwhelmingly by irregularities; set the 1st octant, including the coordinate areas, and the inconsistency - napivprostir, scho to revenge a cob of coordinates (invert)+ the area itself. The "vertical" area of the rosette is parabolic with a parabola and on the armchair bazhano. For the whole, it is necessary to know the preadatk anchor point, or, simply, the vertex of the parabola. (View the value і rozrahovuєmo vіdpovіdne "z").

I will sell it:

Butt 3

Calculate for the help of the waste integral the volume of the body surrounded by the values of the surfaces. Viconati armchair.

Decision: the formula "visonati armchair" gives us freedom, ale, shvidshe for everything, transferring the vison of a spacious armchair.

Dotrimuєmosya іdpratsovanoi earlier tactics - a collection of surfaces, which are parallel to the axis of the applique. Rivnyannya such surfaces do not explicitly take revenge on the "z":

- Rivnyannya set the coordinate area to pass through the line ( yak on the area of the name of the "one-day" rivnyannya);
- івняння back flatness, go through the "same name" "Flat" straight parallel to the axis.

The body, scho to joke, is surrounded by an area below і parabolic cylinder above:

The order of going around the floor is easy, with a lot of "iksov" and "players" between the integration, I guess it’s better to go behind the double armchairs:

In this rank:

When integrating for the "player" - "ix" is used as a constant, that constant is worthily blaming for the integral sign.

View:

So, without forgetting a little, the result is low-grade (and fast), looking with trivial armchairs, a bit of a great experience iliusia obsyagu, About yaku I rozpoviv shche at urotsi Ob'am tila wrapping... So, I appreciated the quietly discernible staff, especially when I was feeling better, that there are more than 4 "cubes" in the new nabagato.

Offensive stock for self-defense:

Butt 4

Calculate for the help of the waste integral the volume of the body surrounded by the values of the surfaces. Make the chair of the given body of that projection onto the square.

Introductory design for a lesson.

It is not frivolous, if the display of a trivial chair is not difficult:

Butt 5

Behind the help of the wasteful integral, to know the volume of the body, given by the surfaces, to surround it.

Decision: the projection here is awkward, ale over the order її I go around the demand to think Yaksho vibrate the 1st way, then the figuru will be divided into 2 parts, so it’s awkwardly cluttering the calculated sumi two third-party integration. There is a different path for the future. The projection of this body on the armchair is visually and visibly visual:

I ask them for the quality of the pictures, I circulate them directly from the authoritative manuscripts.

Vibiraєmo big general procedure for bypassing figuri:

Now on the right behind the floor. Below it is surrounded by an area, above - by an area, yak pass through the hanging ordinates. But everything is nothing, but the rest of the area is steep and the area is not so simple. The vibe here is unenviable: either the jewelry robot is on a different scale (it’s just a little thinner), or an armchair with a height of close to 20 centimeters (the one that can fit in).

Ale є and the third, spontaneously Russian method of solving the problem - hammer =) And the substitute for a trivial armchair is given a verbal description: “Tse tilo surrounded by cylinders і an area to the side, an area from below, and an area from above. "

"Vertical" between the integration, obviously, is as follows:

It is numbered in volume, not to be forgotten, but by projection, we have done it in a broader way:

View:

As they remembered, they were promoted in the houses not for a hundred bucks, often surrounded by an area below. If it’s not, as a rule, you will need to be ready - you can eat the factory, de tilo roztashovani pid area. So, for example, if in the selection of tasks to replace the area, then it was only possible to symmetrically appear in the lower space, and it would be bordered by the area at the bottom, and by the area at the top!

It is easy to perekanatisya, so you will see the very result:

(Pam'yataєmo, tilo need to go around strictly below the hill!)

In addition, the area can be “fond of”, it can be seen not at the right, the simplest butt: a cool, roasted in the area - if you count it, you don’t know it.

All images are understandable, but as of now, they are not similar to the one for an independent version:

Butt 6

For the help of the costly integral to know the volume of the body, surrounded by surfaces

A short solution and a summary of the lesson.

Moving on to another paragraph with not less popular materials:

The third integral at the cylindrical coordinates

Cylindrical coordinates - tse, by day, polar coordinates in the open space.
In a cylindrical coordinate system, the position of a point is defined by the polar coordinates of the point - the projection of the point onto the area of the point itself.

Go from the trivial Cartesian system to the cylindrical coordinate system, use the following formulas:

One hundred percent of our revision of the viglyad is like this:

I, apparently, in the generalized view, what is the view of the statistics:

Smolder, do not forget about the dodatkovy multiplier "er" and place it correctly polarity of integration when bypassing the projection:

Butt 7

Decision: in the same order do: we look ahead to the ryvnyannya, in which every day there is a “zet” There is one here. Projection cylindrical surface on the area є "the same" colo .

Ploshchini flank the shukane only from below і from above ("hanging" from the cylinder) і designed in a number of:

On the back of a trivial armchair. The main difficult fields in the provinces of the area, yak overturned the cylinder before "mowing" the kutom, as long as you go elips... We will clarify the rewrites analytically: for the whole rewritable area of \ u200b \ u200bthe functional view і numbered function ("visota") in points that ask for, which lie on the boundary of the projection:

The point on the armchair is uniquely known and accurately (not like that, yak i =)) from the line:

The projection of the floor onto the area є colo, and the whole argument for the transition to a cylindrical coordinate system:

It is known that the surface is equal to the cylindrical coordinates:

Now the order of the bypassing of the tila has moved.

A collection of free items from the projection. What is the procedure for circumventing? Similarly, yak i at numbered subintegrals at polar coordinates... Here is a wine element:

"Vertical" between the integration is also obvious - it is entered into the body through the area and is entered through the area:

Let's move on to repeated integrals:

At the same time, the multiplier "ep" is immediately put into the "sviy" integral.

Vinik yak zavzhdi lighter than zlamati on twigs:

The result of the offensive Integral is wearable:

And then it’s not forgotten, but fі vvazhaєmo constant. Ale tse before the singing hour:

View:

More information for independent version:

Butt 8

Calculate the volume of the body enclosed by surfaces as a supplementary integral part. Viconati armchair of the given building and projection onto the area.

A glimpse of the final design for a lesson.

To brutalize respect, but in the minds of the zhdan zhodnogo word it is not said about the transition to a cylindrical coordinate system, and there is no need for people to fight with important integrals in Cartesian coordinates. ... And maybe it won't be - even the third, the Russian way of dealing with problems.

All just fix it! ... in a good sense: =)

Butt 9

For the help of the costly integral to know the volume of the body, surrounded by surfaces

Modestly and relish.

Decision: tse tilo obmezhene end surfaceі elephantine parabolic... Readers who respectfully familiarized themselves with the materials of the statistics Main surfaces for space, they also imagined that they could see it tilo, but in practice they often trawl the folds, so I will give a report on the analytical world.

We know a lot of lines, which are overwhelmed by the surface. Warehouse and Virishimo this system:

From the 1st ryvnyannya term by term to each other:

The result is two roots:

By the way, we know the importance of being equal to the system:
viplya stars
Otzhe, the root of the form is one point - a cob of coordinates. Naturally - even the tops of the cichus are on top of them.

Nowadays there is a different root - also in the case of an equal system:

What kind of geometric zm_st of the gained result? "At the height" (near the area), the paraboloid and the cone overhang cola- a single radius centered at the point.

At the same time, the "bowl" of the paraboloid pushes the "funnel" of the cone, that pretend the end surfaces are followed by a dashed line (behind a vignette I will see far away from us, as can be seen from this angle):

Projecting the building onto the area є colo centered on the cob of coordinates of radius 1, which I didn’t try to visualize through the evidence of this fact (protest writing comment robimo!)... Before the speech, at the two workers in front of the chair projection can be hammered, yakby is not smart.

When going to cylindrical coordinates, the standard formulas are used to record the inconsistency in the simplest viewer and in order to bypass the projection of current problems:

It is known that the surface of the cylindrical coordinate systems is equal:

So, if the task is to look at the upper part of the cone, then from the straight bend:

"Skanuєmo tilo" from the bottom up the hill. The lighthouse is entered through eliptic parabolic i go through the end surface. In such a rank, the "vertical" order of bypassing the body:

Іnsha on the right technics:

View:

It’s not fancy, if you just wondered not to flank him with surfaces, but without any inconsistencies:

Butt 10

Geometric zm_st I have made a report on wide-ranging irregularities in the same pre-existing statutes. The main surfaces for space and for.

I want to take advantage of the parameter, but it does not allow the display of the exact chair, but it gives an important view of the floor. Think yak will be a visonati. A short solution and a summary - for example, a lesson.

... well, well, how's it going? Thinking to finish the lesson, ale so and see how you want it =)

Butt 11

For the additional cost of the integral, calculate the volume of the given file:
, De - A rather positive number.

Decision: inconsistency set a quill with the center on the cob of coordinates to the radius, and the inaccuracy - "Inside" of the circular cylinder with the symmetry of the radius. Such a rank, til, how to make noise, is surrounded by circular cylindrome from the side and symmetrical spherical segments above and below the area.

Accept for the base unit vimira, vikonaєmo armchair:

More precisely, the next step is to call it a little bit, some of the proportions along the axis are not even better. In protest, for the sake of justice, behind the mind, there is no need for any kind of armchair and such an illusion has appeared in abundance.

I’m honored, but here it’s not obligatory to hang up, on the cylinder you hang from the “caps” - if you pick up the compasses and put it in your hands with the center on the cob of coordinates to a radius of 2 cm, then the points will overtake the cylinder in your own ...

Let the two straight-line coordinate systems near the open space
, that system of functions

(1)

how to establish a mutually unambiguous relation between points of certain regions
і
have qix coordinate systems. Suppose that the functions of the system (1) may
uninterrupted private logs. Business card holder, warehouses from cich private old

I call them the Jacobian (or the name of the Jacobian) of the systems and functions (1). Mi let it go
v
.

In vicladeni vische poppushennyakh maє mіsce, the zagalnaya formula for replacing the winners with the consumer integration is nascent:

Yak і in the form of a sub-base integral, in exchange for the unambiguity of the system (1) that umova
It can be porous at the edges of the points, the edges of the lines and the edges of the surfaces.

System of functions (1) skin points
put one point at the end
... Three numbers
call the curved coordinates of the point ... Point to the vastness
, if we take one of the three coordinates zberiga permanently value, set the sov. coordinate surface.

II Secondary integral at cylindrical coordinates

The cylindrical coordinate system (CSK) is based on an area
, in which the polar coordinate system is given
, perpendicular to the area. Cylindrical coordinates of a point
, de
- Polar coordinates of the point - Projections t eyepieces on the square
, a - Tse coordinates of the projection point for free
abo
.

Near the area
Introduced by the variable rank of Cartesian coordinates
PPM. Now, it’s not very important to make out the formulas that call the cylindrical coordinates with the Cartesian ones:

(3)

The formulas render the area over the entire space
.

Coordinate surfaces in this view will be:

1)
- cylindrical surfaces with fixed, parallel axes
, straighten how to serve a stake in the area
, centered at points ;

2)

;

3)
- areas, parallel areas
.

Jacobian system (3):

The nagal formula at the same CSK nabuva viglyadu:

Respect 1 . The transition to cylindrical coordinates is recommended if the area of integration is a chain of circles of a cylinder or a cone, or a parabolic wrap (or a part), and the whole area is formed from the application
.

Note 2. Cylindrical coordinates can be displayed as polar coordinates on the area.

stock 1. Calculate the third-party integral from the function

by region
, which is the inner part of the cylinder
, surrounded by a cone
that parabolic
.

Decision. The qiu region was also looked at in §2, butt 6, and the standard DPSK record was taken away. Protest, the calculation of the integral is important. Let's move on to CSK:

Projection
tila
on the square
- tse colo
... Otzhe, coordinate change from 0 to
, a - from0 to R. Through a certain point
carried out straight, parallel to the axis
... Go straight to
on a cone, and on a paraboloid. Ale cone
maє u CSK rivnyannya
, and paraboloid
- Rivnyannya
... Otzhe, maєmo

III Secondary integral at spherical coordinates

Spherical coordinate system (SSC) is
, in which the UCS is set, that vissyu
perpendicular to the area
.

Spherical point coordinates give space to three numbers
, de - Polar kut of the projection of the point on the area
,- Кут між віссю
that vector
і
.

Near the area
introduced Cartesian coordinate axes
і
by the highest rank, and the aplikat of the summit was
... The formulas for tying spherical coordinates with Cartesian ones are as follows:

(4)

The formulas render the area over the entire space
.

Jacobian systems and functions (4):

Coordinate surfaces have three families:

1)
- Concentric spheres centered on the cob of coordinates;

2)
- napіvploshini, scho pass through the rope
;

3)
- Circular cones with a top on the cob of coordinates, which serve to hang
.

The formula for the transition to SSK at the consumer integration:

Note 3. It is recommended to go to the SBK if the integration area is a tse kulya abo її part. At the same level of spheres
go to. Yak i CSK, viewed earlier, SSK is "tied" to the axis
... As the center of the spherical adjustment on the radius of the coordinate axis
:

Respect 4. Can you use the SBK:

with the Jacobian
... Qia system of functions translate e-mail

in "parallelepiped"

stock 2. Know the middle, get the points of the cooler radius go to the center.

Decision. Nagadaєmo, the middle value of the function
in the area
- a wide range of integral part of the function of the region of the region. Our vipadku