44 physical additions to the singing integral inurl livre. Geometric definition of the value of the integral. Obsyag tila wrapping

41.1. Integral storage schemes

Do not need to know the meaning of a geometrical or physical size A (area of ​​a figuri, volume of til, a grip of a line on a vertical plate, etc.), tied from the general serpent of an independent winter. Transferred, well, the value of A is additive, that is, E. Such, when rozbittі vіdrіzka [a; b] by the point h є (a; b) on the part [a; h] i [s; b] the value of the value A, in all cases as [a; b], dorіvnyu sumі її meaning, scho іdpovіdaut [a; h] i [s; b].

For the knowledge of the value of A, it is possible to keruvatise one of two schemes: Scheme I (or the method of integral sums) and II scheme (or the differential method).

The first scheme is based on the designation of the singing integral.

1. Dots x 0 = a, x 1, ..., x n = b to break out into [a; b] into n parts. As a matter of fact, the value of A will rise to n "elementary supplements" ΔAi (i = 1, ..., n): А = ΔA 1 + ΔА 2 + ... + ΔА n.

2. Demonstration of the skin "elementary dodanok" in the view of the creation of the deyakoi funktsii (as it should start from the mind of the tasks), calculated in the most important point of the appropriate result for the yogi endurance: ΔA i ≈ ƒ (c i) Δx i.

With a known close value of ΔA i, let us assume that an arc is forgiven: an arc on a small delay can be replaced by a chord, which is pulled together; The change in speed on a small date can be made very quickly, etc.

Otrimamo close to the value of A in the integral sumi:

3. Shukan is the value of A before the boundary of the integral sumi, i.e. E.

The meanings "method of sums", like bacimo, are used in the presentation of the integral, like about the sum of an infinitely great number of infinitely small numbers.

Scheme I of the ball is stuck for the setting of a geometric and physical zmisty singing integral.

Another scheme is just a modification of scheme I and is called the "differential method" or the "method of seeing indefinitely small different orders":

1) on vidrizka [a; b] vibramally conservative values ​​of x and display of changes in the display [a; NS]. On the whole, the value of A becomes a function of x: A = A (x), i.e., E. Vvazhamo, which is a part of the shukan value of A є is not a function of A (x), de x є is one of the parameters of the value of A;

2) we know the head part of the increment ΔA when changing x by a small value Δx = dx, i.e., the differential dA of the function A = A (x) is known: dA = ƒ (x) dx, de tasks, functions of change (here you can also ask for help);

3) vvazayuchi, scho dA ≈ ΔA at Δх → 0, the shukan knows the value of integration dA in the intervals from a to b:

41.2. Calculating the area of ​​flat figures

rectangular coordinates

As it is already established (div. "Geometric sensory integral"), the area of ​​the curved trapezium, the rosted "vishche" of the abscis axis (ƒ (x) ≥ 0), back to the common singing integral:

Formula (41.1) is rimmed by way of storing scheme I - the sum method. Formula (41.1), vicorist scheme II. Let the curved trapezium be surrounded by lines y = ƒ (x) ≥ 0, x = a, x = b, y = 0 (div. Fig. 174).

For the well-known area of ​​the S trapezium of the vicon of the onset of the operation:

1. Vіzmemo dovіlne x Î [a; b] and we will assume that S = S (x).

2. Damo the argument x pririst Δx = dx (x + Δx є [a; b]). The function S = S (x) is to get an increase of ΔS, which is an area of ​​an "elementary curved trapezium" (a tiny picture of it).

Differential area dS є head part of the increment ΔS at Δх → 0, і, obviously, in the road area of ​​the rectangle with the base dx and the height y: dS = y dx.

3.Integrating the differentiation of equality in the boundaries from x = a to x = b, obsessed

Obviously, the curvilinear trapezoid is rounded "below" the axis Ox (ƒ (x)< 0), то ее площадь может быть найдена по формуле

Formulas (41.1) and (41.2) can be combined into one:

The area of ​​the figuri, surrounded by curves y = = fι (x) і у = ƒг (x), by lines x = a і х = b (behind ƒ 2 (x) ≥ ƒ 1 (x)) (div. Fig. 175) , you can know behind the formula

If the figure is flat, I have a "folding" shape (div. Fig. 176), then straight, parallel to the axis Oy, and then cut into pieces, so that you can use the same formulas.

As the curved trapezium is surrounded by straight lines y = s і y = d, vіssu Oy і uninterruptedly curved х = φ (y) ≥ 0 (div. Fig. 177), then the її area is behind the formula

I, nareshty, like a curvilinear trapezoid surrounded by a curve, given parametrically

straight lines x = aih = bі vіssu Oh, then the area її is behind the formula

de a and β are determined from the equivalence x (a) = a і x (β) = b.

Butt 41.1. To know the area of ​​the figuri, surrounded by the vissu Oh and by the graph of the function y = x 2 - 2x at x.

Solution: Figura maє viglyad, images on baby 178. It is known in the area S:

Butt 41.2. Count the area of ​​the figurine, surrounded by an ellipse x = a cos t, y = b sin t.

Decision: It is known from a group of 1/4 area S. Here x changes from 0 to a, from the same, t changes from 0 to 0 (div. Fig. 179). it is known:

In such a rank. Hence, S = π аВ.

polar coordinates

We know the area S of the curved sector, i.e., a flat figure, interlaced with an uninterrupted line r = r (φ) and two interchanges φ = a і φ = β (a< β), где r и φ - полярные координаты (см. рис. 180). Для решения задачи используем схему II - differential method.

1. Let us take a part of the Shukan area S as the function of the kut φ, that is, S = S (φ), if a ≤ φ ≤ β (if φ = a, then S (a) = 0, if φ = β, then S (β) = S).

2. If the current polar cut φ gains an increase of Δφ = dφ, then the AS area will increase to the area of ​​the “elementary curved sector” OAB.

Differential dS є the head part of the increment ΔS at dφ → 0 і road areas of the circular sector About the AC (shaded per minute) of the radius r with the central edge dφ. Tom

3.Integrating the alignment between the boundaries from φ = a to φ = β,

Butt 41.3. Know the area of ​​the figuri, surrounded by the "three-petal trojan" r = acos3φ (div. Fig. 181).

Decision: It is known that the area of ​​half of one "Trojandi" peel is known, that is, 1/6 of the whole area of ​​the figurine:

that is. Otzhe,

As the figure is flat, I “fold” the shape, then in turns, going out from the poles, and going to the curved sectors, until I’m fixing the formula for the known area. So, for a figuri, an image on a baby 182, maєmo:

41.3. Calculation of the arc of a flat crooked

rectangular coordinates

Let in straight-line coordinates a flat curve AB is given, equal to (x), de a≤x≤ b.

From the line of the arc AB, the boundary grows, until it is pragmatic until the laman line is inscribed in the arc, if the number of laman lanes is not between growth, and the number of laman lines is not equal to zero. It will be shown that if the function y = ƒ (x) і її is inherited from "= ƒ" (x) is not interrupted by the way [a; b], then the curve AV is maє dovzhinu, rіvnu

Zastosuєmo scheme I (sum method).

1.Dots x 0 = a, x 1 ..., x n = b (x 0< x 1 < ...< х n) разобьем отрезок [а; b] на n частей (см. рис. 183). Пустьэтим точкам соответствуют точки М 0 = А, M 1 ,...,M n =В накривой АВ. Проведем хорды М 0 M 1 , M 1 M 2 ,..., М n-1 М n , длины которых обозначим соответственно через ΔL 1 , AL 2 ,..., ΔL n . Получим ломаную M 0 M 1 M 2 ... M n-ι M n , длина которой равна L n =ΔL 1 + ΔL 2 +...+ ΔL n =

2. Dovzhin jordi (Abolanka Lamano) ΔL 1 can be known for the theorem of Pythagoras with a tricot with legs Δx i і Δу i:

According to Lagrange's theorem about the kintsevy extra functions Δу i = ƒ "(з i) Δх i, de ci є (x i-1; x i).

and all the lamanos are allowed to eat M 0 M 1 ... M n dorivnyu

3.Lina l crooked AB, for viznachennyam, dorivnyu

.

Remarkable, when ΔL i → 0 also і Δx i → 0 ΔLi = i, also, | Δx i |<ΔL i).

function uninterrupted for the duration of [a; b], so that, behind the sink, the function ƒ "(x) is not interrupted. Otzhe, іnuє between the integral sumi (41.4), if max Δx i → 0 :

In such a rank, but in a fast recording l =

If the equation of the curve AB is given in the parametric form

de x (t) і y (t) - continuous functions with uninterrupted functions і х (а) = а, х (β) = b, then dovzhina l crooked AV is behind the formula

Formula (41.5) can be trimmed from formula (41.3) by setting x = x (t), dx = x "(t) dt,

Butt 41.4. Know the dinner of a stake of Radius R.

Solution: We know 1/4 part of the її dozhini from point (0; R) to point (R; 0) (div. Fig. 184). So yak then

to mean, l= 2π R. If you write a stake in the parametric view x = Rcost, y = Rsint (0≤t≤2π), then

The calculation of the arc can be based on the differential method. It will be shown that it is possible to reject formula (41.3), having stagnated scheme II (differential method).

1. The value of x є [a; b] і clearly visible display [a; NS]. New value l a new function from x, tobto l = l(NS) ( l(A) = 0 і l(B) = l).

2. We know the differential dl functions l = l(X) when changing x by a small value Δx = dx: dl = l"(X) dx. We know l"(X), replace the infinite-small arc MN with the chord Δ l, Contracting qiu arc (div. Fig. 185):

3.Integrates dl between from a to b, obsessed

parity is called the arc differential formula in straight-line coordinates.

So yak y "x = -dy / dx, then

Remaining the formula є Pythagoras' theorem for infinitely small tricycle MST (div. Fig. 186).

polar coordinates

Let the AB curve be set equal in polar coordinates r = r (φ), and≤φ≤β. It is admitted that r (φ) і r "(φ) is not interrupted by the direction [a; β].

If in the equalities x = rcosφ, y = rsinφ, where the polar and Cartesian coordinates are used, the parameter is equal to φ, then the AB curve can be set parametrically

Zastosovuchi formula (41.5),

Butt 41.5. Know the amount of cardioid r = = a (1 + cosφ).

Solution: Cardioid r = a (1 + cosφ) ma viglyad, images on a baby 187. Vona is symmetrical to the polar axis. We know half of the total amount of cardioidi:

In this rank, 1 / 2l = 4a. Hence, l = 8а.

41.4. Calculated obsyagu tila

The calculation of the amount of money for each area of ​​parallel perereziv

Do not need to know the volume of the V floor, moreover, in the area S of the cross-section of the floor, the areas perpendicular to the main axis, for example, the Ox axis: S = S (x), a ≤ x ≤ b.

1. Through a sufficient point x є draw a plane Π, perpendicular to the axis Ox (div. Fig. 188). In terms of S (x), the area is overrun by a whole area; S (x) vazhaєmo see and change without interruption when changing. Through v (x) it is meaningful to obsyag part of the body, how to lie more than the area P. x] the value v is the function of x, that is, v = v (x) (v (a) = 0, v (b) = V).

2. We know the differential dV of the function v = v (x). Win is an "elementary ball" of the floor, laying between parallel areas, which overshadows Ox at the points x і х + Δх, which can be approximately taken over the cylinder with the base S (x) і height dx. To that the differential volume dV = S (x) dx.

3. Known to the shukan the value of V by the path of integration dA in the boundaries from a to B:

Otriman formula is called the formula of obsyag tila on the area of ​​parallel crossings.

Butt 41 .6. Know obsyag elipsoyda

Solution: Rozsіkayuchi elіpsoїd area, parallel area Oyz and in the first place ≤х≤ a), otrimaєmo elips (div. fig. 189):

The area of ​​\ u200b \ u200bth ellipse

Tom, by formula (41.6),

Obsyag tila wrapping

Do not go near the axis Oh to wrap a curved trapezium, surrounded by an uninterrupted line y = ƒ (x) 0, with a long a ≤ x ≤ bі by straight lines x = a і x = b (div. Fig. 190). Otrimana from the wrapping of the figure is called the wrap-around. Peretin ts'go til with an area perpendicular to the Ox axis, drawn through a certain point x of the Ox axis (x Î [A; b]), є colo with radius у = ƒ (x). From the same, S (x) = π y 2.

Zastosovyuchi formula (41.6) obsyagu til on the area of ​​parallel crossings, can be recognized

As a curved trapezium is surrounded by a graph not without interruption of the function = φ (y) ≥ 0 and straight lines x = 0, y = c,

y = d (s< d), то объем тела, образованного вращением этой трапеции вокруг оси Оу, по аналогии с формулой (41.7), равен

Butt 41.7. To know the volume of the til, set to the wraps of the figuri, surrounded by the lines around the axis Oy (div. Fig. 191).

Decision: For the formula (41.8) we know:

41.5. Estimated surface area wrapping

Let the curve AB є graph function y = ƒ (x) ≥ 0, de x є [a; b], and the function y = ƒ (x) і її is inherited from "= ƒ" (x) is not interrupted in any way.

We know the area S of the surface, set to the wraps of the curved AB near the axis Ox.

Zastosuєmo scheme II (differential method).

1. Through a certain point x є [a; b] draw an area Π perpendicular to the Ox axis. The area Π overflows the surface of the wrap around the stake with radius y = ƒ (x) (div. Fig. 192). The value S of the surface part of the figure is wrapped, but to lie more than the area, is a function of x, i.e. S = s (x) (s (a) = 0 і s (b) = S).

2. Damo argument x pririst Δx = dx. Through the point x + dx є [a; b] a plane perpendicular to the Ox axis is also drawn. The function s = s (x) is to get the increase of Az, which is pictured on the little one by the "Pask" viewer.

We know the differential of the area ds, I will fix it with the grips of the figure, we will increase the cone, I will fix it. dl, A radiusi is equal to y + dy. The area of ​​the second surface of the road ds = π (Y + y + dy) dl=2π at dl + π dydl... Open TV dydl as an infinitely small order, lower ds, acceptable ds = 2 π at dl, Abo, so yak

3.Integrating the differentiation of equality in the boundaries from x = a to x = b, obsessed

If the AB curve is given by parametric equivalents x = x (t), y = y (t), t 1 ≤ t ≤ t 2, then the formula (41.9) for the area of ​​the wrap-around surface

Butt 41.8. Know the area of ​​the surface of the cooler of the radius R.

Butt 41.9. given cycloid

To know the area of ​​the surface, set to the wraps around the axis Oh.

Solution: When half of the arc is wrapped, the cycloid is about the Ox axis, the surface area is wrapped.

41.6. Mechanical additions of the singing integral

Robot of wintry strength

Let the material point M move bridging the axis Oh before the change of force F = F (x), aligned parallel to the axis. A robot, vibrated by force when the point M is displaced from the position x = a in the position x = b (a< b), находится по формуле (см. п. 36).

Butt 41.10 Yak robot needs to spend, to stretch the spring by 0.05 m, if the force is 100 N to stretch the spring by 0.01 m?

Decision: Behind Hooke's law, the spring force, which stretches the spring, is proportional to the stretching x, i.e. F = KX, de k is the coefficient of proportion. At the end of the washing task, the force F = 100 N pulls the spring to x = 0.01 m; the same, 100 = k * 0.01, stars k = 10000; the same, F = 10000x.

Shukana of the robot on the basis of the formula (41.10)

Butt 41.11. To know to the robot, if it is necessary to spend it, to wickachati over the edge of the groove from the vertical cylindrical reservoir with the height H m and the radius of the base R m.

Solution: A robot that is able to turn on the height of the height of the height h, road to the height of the h. Alle the growth balls in the tanks are located on the lower slopes and the height of the lift (to the edge of the reservoir) of the small balls is not the same.

Scheme II (differential method) is used for the verification of the set designation. The coordinate system is introduced as indicated on little 193.

1. Robot, scho to see the vikachuvannya from the tank ball ridini tovshchinoyu x (0 !!!< x !!!< H), есть функция от х, т.е. А = А(х), где 0≤x≤H (А(0)=0, А(Н)=А 0).

2. It is known the head part of the increment ΔA when changing x by the value Δx = dx, ie. The differential dA of the function A (x) is known.

Zvazhayuchi on krykhta dx vazhaєmo, so the "elementary" ball of the line is located on one gauge (towards the edge of the reservoir) (div. Fig. 193). Todi dA \ u003d dp * x, de dp - wha ts'go ball; vіn dorіvnyu g * g dv, de g - accelerated vіnnogo fadіnnya, g - proficiency of rіdini, dv - obsyag of the "elementary" ball of rіdini (for a little wіn visions), that is, dp = gg dv. Obshy to the designated ball rіdini, obviously, dorіvnyuє π R 2 dx, de dx - the height of the cylinder (ball), π R 2 - the area of ​​your sleep, tobto. E. Dv = π R 2 dx.

In such a rank, dp = gg π R 2 dx і dA = gg π R 2 dx * x.

3) integrating the trimming of the parity in the boundaries from x = 0 to x = H, it is known

Shlyakh, passages by til

Let the material point move along the straight line through the changing speed v = v (t). We know the way S, it passes for an hour from t 1 to t 2.

Decision: From the physical change of the simple-minded view, from the hour to the point in one straight line, “the speed of the straightforwardness of the straight line to the simple way to the road by the hour”, that is. Integration of the difference between the boundaries from t 1 to t 2, it is recognized

Obviously, the formula can be rejected by using the scheme I or II storing the singing integral.

Butt 41.12. Know the way, passing through in 4 seconds to the ear of the corn, as the speed of the floor is v (t) = 10t + 2 (m / s).

Decision: If v (t) = 10t + 2 (m / s), then a walk, only passes through the ear of a corn (t = 0) until the end of the 4th second, road

Ridini vise on a vertical plate

Obviously, due to Pascal's law, the grip of a line on a horizontal plate is an early stage of a line of a line, when I pay a fee, and by weight - a depth of line from the vertical surface of the Sidin, i.e., E. P * = h * de g * floor, g - thickness of the line, S - area of ​​the plate, h - surface area.

For this formula, you can shukati the grip of the line on the vertically bored plate, so that the point lies on the small slopes.

Let the plate be bored vertically into the road, surrounded by lines x = a, x = b, y 1 = f 1 (x) і y 2 = ƒ 2 (x); the vibran coordinate system is so, as indicated on the little one 194. For the knowledge of the grip of the Ridini on the plate, the scheme II is used (the differential method).

1. Let the part of the shukanoi value P є function from x: p = p (x), ie P = p (x) - the vice on the part of the plate, like the one [a; x] the value of the wrinkle x, de x є [a; b] (p (a) = 0, p (b) = P).

2. Damo argument x pririst Δx = dx. Function p (x) to win? P (for a baby - a small ball of dx). We know the differential dp of the function. Rattling on the dx krykhta, we will be close to the square with a rectangle, all the specks of which are found on the same glybin, that is, the plate is horizontal.

Todi behind Pascal's law

3.Integrating the trimming of the parity in the boundaries from x = a to x = B,

Butt 41.13. Viznachit the size of the grip of the drive on the wheel, vertically in the path, where the radius is R, and the center Pass on the surface of the drive (div. Fig. 195).

The static moment S y of the system of the axis

As the masi rozpodіlenі bezperervnim rank of bridle deyakoi crooked, then for the rotation of the static moment, integrate.

Nekhai y = ƒ (x) (a≤ x≤ b) - the value of the material curve AB. We will vvvat її one-sided with a post-lineal line g (g = const).

For a previlny x є [a; b] on the AB curve there is a point with coordinates (x; y). Visible on the curve of the elementary dl, to take revenge on the point (x; y). Todi masa tsієї dilyanka dorіvnyu g dl. Acceptable dl is close to the point, from the distance from the axis Oh to the back. The differential of the static moment dS x ("elementary moment") will be suitable for g dly, ie DS x = g dlу (div. Fig. 196).

Svidsy vyplyaє, but the static moment S x curved AB from the axle Oh dorіvnyuє

Similarly, we know S y:

Static moments S x і S y crooked make it easy to set the position of the center of the vagi (center of the mass).

The center of the heavy material flat curve y = ƒ (x), x Î is the point of the area, when Volodya is the offensive power: if in the whole point of the middle the entire mass m is given crooked, then the static moment of the road of the point is, as a coordinate crooked y \ u003d ƒ (x) is very similar to the axis. Let us denote by C (x c; y c) the center of the vagi of the curve AB.

The center of the car should be equal Zvidsi

Calculation of static moments and coordinates of the center of a wagi of a flat figure

Let there be given a material flat figure (plate), surrounded by a curve y = ƒ (x) 0 and straight lines y = 0, x = a, x = b (div. Fig. 198).

We will take into account that the surface area of ​​the plate is permanent (g = const). Todi masa "all plates are doors g * S, i.e. E Visible elementary delink of the plate near the viglyad, indefinitely high vertical smog, and will be approached by a straight-forward.

Todi masa yogo dorivnyuє g ydx. The center of gravity of the Z rectangle lies on the cross-section of the diagonals of the rectangle. The point C goes from the axis Ox to 1/2 * y, and from the axis Oy to x (close; more precisely, at the point of x + 1/2 Δx). Todi for the elementary static moments of the axes Oh and Oy

Otzhe, center of wagi maє coordinates

The values ​​of integral (OI) are widely used in practical additions to mathematics and physics.

In the wake of the day, in the geometries behind the other OI there are areas of simple figures and folding surfaces, volumetric wraparound and modern shape, more curves on the area and in space.

Physics and theoretical mechanics of OI are used for calculating static moments, mass and centers of masses of material curves and surfaces, for calculating robotic force along a curved path and in.

The area of ​​the flat figuri

Do not have a flat figure in the Cartesian rectangular coordinate system $ xOy $ at the top surrounded by a curve $ y = y_ (1) \ left (x \ right) $, at the bottom - by a curve $ y = y_ (2) \ left (x \ right) $, and on the right side by vertical lines $ x = a $ і $ x = b $ apparently. In a zealous vipad area of ​​such a figure, turn around for an additional OI $ S = \ int \ limits _ (a) ^ (b) \ left (y_ (1) \ left (x \ right) -y_ (2) \ left (x \ right ) \ right) \ cdot dx $.

Also, a flat figure in the Cartesian rectangular coordinate system $ xOy $ is surrounded by a curve $ x = x_ (1) \ left (y \ right) $ on the right, a curve $ x = x_ (2) \ left (y \ right) $ , and below and above by horizontal straight lines $ y = c $ і $ y = d $ as if, then the area of ​​such a figure will turn behind the other OI $ S = \ int \ limits _ (c) ^ (d) \ left (x_ (1) \ left (y \ right) -x_ (2) \ left (y \ right) \ right) \ cdot dy $.

Do not have a flat figure (vignute sector), which can be viewed in polar coordinate systems, is set by the graph of continuous function $ \ rho = \ rho \ left (\ phi \ right) $, as well as two interchanges to go through $ \ phi = \ alpha $ i $ \ phi = \ beta $ is correct. The formula for calculating the area of ​​such a curved sector of the ma viglyad: $ S = \ frac (1) (2) \ cdot \ int \ limits _ (\ alpha) ^ (\ beta) \ rho ^ (2) \ left (\ phi \ right ) \ cdot d \ phi $.

Dovzhina arc crooked

$ \ Left [\ alpha, \; \ Beta \ right] $ the curve is set equal to $ \ rho = \ rho \ left (\ phi \ right) $ in polar coordinate systems, then the arc of the arc is calculated according to the additional OI $ L = \ int \ limits _ (\ alpha) ^ (\ beta) \ sqrt (\ rho ^ (2) \ left (\ phi \ right) + \ rho "^ (2) \ left (\ phi \ right)) \ cdot d \ phi $.

If the curve is given equal to $ y = y \ left (x \ right) $, then the curve of the arc is calculated for the additional OI $ L = \ int \ limits _ (a) ^ (b) \ sqrt (1 + y "^ (2) \ left (x \ right)) \ cdot dx $.

$ \ Left [\ alpha, \; \ Beta \ right] $ the curve is given parametrically, so that $ x = x \ left (t \ right) $, $ y = y \ left (t \ right) $, then the arc її is calculated for the additional OI $ L = \ int \ limits _ (\ alpha) ^ (\ beta) \ sqrt (x "^ (2) \ left (t \ right) + y" ^ (2) \ left (t \ right)) \ cdot dt $.

Enumeration of the obsyagu tila behind the areas of parallel perereziv

Do not need to know the volume of the spacious floor, the coordinates of the points of which we are happy with $ a \ le x \ le b $, and for which in every area there is a cross in $ S \ left (x \ right) $ with areas perpendicular to the axis $ Ox $.

The formula for calculating such a tila maє viglyad is $ V = \ int \ limits _ (a) ^ (b) S \ left (x \ right) \ cdot dx $.

Obsyag tila wrapping

Let's go to $ \ left $, a nonnegative non-intermittent function $ y = y \ left (x \ right) $ is given, which creates a curved trapezoid (CRT). If you wrap the MCT around the $ Ox $ axis, then you pretend to be just, called by the wrapping.

Numeric obsyagu tila wrapping є we will limit the number of numbered tila behind the given areas of parallel transitions. Like the formula of the maverick $ V = \ int \ limits _ (a) ^ (b) S \ left (x \ right) \ cdot dx = \ pi \ cdot \ int \ limits _ (a) ^ (b) y ^ ( 2) \ left (x \ right) \ cdot dx $.

Do not have a flat figure in the Cartesian rectangular coordinate system $ xOy $ at the top surrounded by a curve $ y = y_ (1) \ left (x \ right) $, at the bottom - by a curve $ y = y_ (2) \ left (x \ right) $, de $ y_ (1) \ left (x \ right) $ і $ y_ (2) \ left (x \ right) $ - no function without interruption, and wrong and right vertical lines $ x = a $ і $ x = b $ for sure. Todi obsyag til, adopted by the wraps of the figure of the figure around the axis $ Ox $, turn OI $ V = \ pi \ cdot \ int \ limits _ (a) ^ (b) \ left (y_ (1) ^ (2) \ left (x \ right) -y_ (2) ^ (2) \ left (x \ right) \ right) \ cdot dx $.

Do not have a flat figure in the Cartesian rectangular coordinate system $ xOy $ on the right is surrounded by a curve $ x = x_ (1) \ left (y \ right) $, in the wrong - a curve $ x = x_ (2) \ left (y \ right) $, de $ x_ (1) \ left (y \ right) $ і $ x_ (2) \ left (y \ right) $ - no function without interruption, and below and above by horizontal lines $ y = c $ і $ y = d $ for sure. Todi obsyag til, endowed with the wraps of the figuri around the axis $ Oy $, turn OI $ V = \ pi \ cdot \ int \ limits _ (c) ^ (d) \ left (x_ (1) ^ (2) \ left (y \ right) -x_ (2) ^ (2) \ left (y \ right) \ right) \ cdot dy $.

The area of ​​the surface is wrapped

Let's go to $ \ left $ a nonnegative function $ y = y \ left (x \ right) $ with an uninterrupted simple $ y "\ left (x \ right) $ is given. $, then it itself is set to just wrapping, and the arc of the MCT is to its surface. \ right) \ cdot \ sqrt (1 + y "^ (2) \ left (x \ right)) \ cdot dx $.

It is admitted that the curve $ x = \ phi \ left (y \ right) $, de $ \ phi \ left (y \ right) $ - is given to $ c \ le y \ le d $ is a non-negative function, wrap around the axis $ Oy $. At the end of the range of the area of ​​the surface of the set body, the wrapping is twisted OI $ Q = 2 \ cdot \ pi \ cdot \ int \ limits _ (c) ^ (d) \ phi \ left (y \ right) \ cdot \ sqrt (1+ \ phi "^ (2) \ left (y \ right)) \ cdot dy $.

Physical supplements OI

1. For the distance detector, at the time of the hour $ t = T $ with a change in the fluidity of the material point $ v = v \ left (t \ right) $ of the material point, when the drop starts at the time of the hour $ t = t_ (0) $, the result is OI $ S = \ int \ limits _ (t_ (0)) ^ (T) v \ left (t \ right) \ cdot dt $.
2. To calculate the robotic force, $ F = F \ left (x \ right) $, to reach the material point, to move behind the straight line from the $ Ox $ axis from the point $ x = a $ to the point $ x = b $ (directly dії power to get out of the way) vikoristovuyu OI $ A = \ int \ limits _ (a) ^ (b) F \ left (x \ right) \ cdot dx $.
3. Static moments from the coordinate axes of the material curve $ y = y \ left (x \ right) $ to the interval $ \ left $ rotate by the formulas $ M_ (x) = \ rho \ cdot \ int \ limits _ (a) ^ (b) y \ left (x \ right) \ cdot \ sqrt (1 + y "^ (2) \ left (x \ right)) \ cdot dx $ і $ M_ (y) = \ rho \ cdot \ int \ limits _ (a ) ^ (b) x \ cdot \ sqrt (1 + y "^ (2) \ left (x \ right)) \ cdot dx $.
4. The center of a material crooked is a point, in which the whole of the world is cleverly sorted in such a rank that the static moments of the point along the coordinate axes are adjusted to the general static moments of all the crooked ones as a whole.

Formulas for calculating the coordinates to the center of a flat curved mass $ x_ (C) = \ frac (\ int \ limits _ (a) ^ (b) x \ cdot \ sqrt (1 + y "^ (2) \ left (x \ right)) \ cdot dx) (\ int \ limits _ (a) ^ (b) \ sqrt (1 + y "^ (2) \ left (x \ right)) \ cdot dx) $ і $ y_ (C) = \ frac (\ int \ limits _ (a) ^ (b) y \ left (x \ right) \ cdot \ sqrt (1 + y "^ (2) \ left (x \ right)) \ cdot dx) ( \ int \ limits _ (a) ^ (b) \ sqrt (1 + y "^ (2) \ left (x \ right)) \ cdot dx) $.

5. Static moments of a material flat figure at the CMT viewer with a number of coordinate axes rotate by the formulas $ M_ (x) = \ frac (1) (2) \ cdot \ rho \ cdot \ int \ limits _ (a) ^ (b) y ^ (2) \ left (x \ right) \ cdot dx $ і $ M_ (y) = \ rho \ cdot \ int \ limits _ (a) ^ (b) x \ cdot y \ left (x \ right) \ cdot dx $.
6. Coordinate the center of a massive flat figure at the viewer of the MCT, set by the curve $ y = y \ left (x \ right) $ to the interval $ \ left $, calculated according to the formulas $ x_ (C) = \ frac (\ int \ limits _ (a ) ^ (b) x \ cdot y \ left (x \ right) \ cdot dx) (\ int \ limits _ (a) ^ (b) y \ left (x \ right) \ cdot dx) $ і $ y_ ( C) = \ frac (\ frac (1) (2) \ cdot \ int \ limits _ (a) ^ (b) y ^ (2) \ left (x \ right) \ cdot dx) (\ int \ limits _ (a) ^ (b) y \ left (x \ right) \ cdot dx) $.

Topic 6.10. Geometric and physical additions to the singing integral

1. The area of ​​the curved trapezium, interlaced by the curve y = f (x) (f (x)> 0), by the straight lines x = a, x = b and the parallel [a, b] axis Ox, calculated by the formula

2. The area of ​​the figuri, surrounded by curves y = f (x) і y = g (x) (f (x)< g (x)) и прямыми х= a , x = b , находится по формуле

3. If the curve is given by parametric equal parameters x = x (t), y = y (t), then the area of ​​the curved trapezoid, which is surrounded by a straight curve and by the straight lines x = a, x = b, is located behind the formula

4. Nekhai S (x) - the area of ​​the floor is perpendicular to the axis Ox, only the part of the floor, laid down between the perpendicular axis areas x = a і x = b, is located behind the formula

5. Do not go curved trapezium, surrounded by a curve y = f (x) і straight lines y = 0, x = a і х = b, wrap around the axis Oh, todіg the wraparound to be calculated according to the formula

6. Don't go curved trapezium, surrounded by a curve х = g (y) і

straight lines x = 0, y = c і y = d, wrap around the axis O y, todі wrap around the wraparound to be calculated according to the formula

7. If a flat curve is brought to a rectangular coordinate system and is given equal to y = f (x) (or x = F (y)), then the arc gain is set by the formula

Головна> Lectures

Lecture 18. Complements of the singing integral.

18.1. Enumeration of areas of flat figures.

Seemingly, a singing integral on the edge of the area of ​​the curved trapezium, surrounded by a graph of the function f (x). If the graph of sewing is lower than the axis of the Ox, tobto f (x)< 0, то площадь имеет знак “-“, если график расположен выше оси Ох, т.е. f(x) >0, then the area is marked "+".

For the knowledge of the total area, the formula is victorious.

The area of ​​the figurines, surrounded by deyakim lines, can be known behind the help of singing integrals, as well as from the common lines.

Butt. Know the area of ​​the figurines, surrounded by lines y = x, y = x 2, x = 2.

Shukana area (shaded in the figure) can be found behind the formula:

18.2. Knowledge of the area of ​​the crooked sector.

For the known area of ​​the curved sector, a polar coordinate system is introduced. Rivnyannya crooked, which will intertwine the sector in the whole coordinate system, ma viglyad  = f (), de  - dovzhyna radius - vectors, but the one pole from the next point of the curve, and  - kut nahila radius - the vector to the polar ...

The area of ​​the curved sector can be found behind the formula

18.3. The calculation of the curve is crooked.

y y = f (x)

S i y i

Dovzhina lamanoi linea, yaka vidpovidak duzi, maybe you know yak
.

Todi dovzhina arc dorivnyu
.

Three geometric mirkuvan:

At the same hour

Todi can be shown

Tobto

If the curve is given parametrically, then, based on the rules for calculating the old parametrically given, it is

,

de x =  (t) і у =  (t).

what is given spacious curve, І х =  (t), у =  (t) і z = Z (t), then

Yakscho curve is set in polar coordinates, then

,  = f ().

butt: Know the amount of stake given to the family x 2 + y 2 = r 2.

1 way Vislovimo from rіvnyannya zminnu.

I know I'll go

Todi S = 2r. Otrimalnovydom formula dozhini cola.

2 way If you are given a line in polar coordinate systems, then it is obsessed: r 2 cos 2  + r 2 sin 2  = r 2, so that the function  = f () = r,
Todi

18.4. Calculated volume

The calculation of the obsyagu tila behind the various areas of the parallel perereziv.

Let’s go tilo obsyag V. The area of ​​any transverse recession of the til Q, in the form of an uninterrupted function Q = Q (x). Rozib'єmo tilo on the "ball" with transverse crossings, which pass through the points x i rozbittya vіdrizka. Oscillations for some intermediate type of function Q (x) is not interrupted, then it is accepted for the newest one for the least value. Significantly, їх is derived from M i і m i.

If on the cich the most and the smallest overturns if the cylinders are built with parallel axes, then the swings of the cylinders will be similar to each other M i x i i m i x i here i - x i = x.

Provided such encouragement for all types of rosbitty, recognition of cylinders, requests of such parties as per
і
.

When pragmatic to zero, crocus rosbitta , tsi sumi may cause a zagalny border:

In such a rank, the obsyag tila can be of knowledge behind the formula:

In a small number of formulas, those that are necessary for knowledge of the function Q (x) are necessary for the knowledge of the function, but it is problematic for the foldable bodies.

butt: Know about 'um kuli radius R.

At the transverse cross-flaps of the coul, there is a cola of a changeable radius. At the same time from the stream coordinates x tsei radius, follow the formula
.

Todi function of the area of ​​overretin the ma viglyad: Q (x) =
.

Otrimuєmo ob'єm kuli:

butt: To know about the large size of the square of the space S.

When overturned by areas perpendicular to the height, in the course of a period of time, we can see figurians, some of them. Coefficients for the needs of these figures for transportation x / H, de x - go from the area to the top of the pyramid.

Geometry of the view, showing the area of ​​additional figures for the transportation of the facilities in the square, tobto

We will be able to recognize the function of the areas of the retinue:

It is known about the obsyag of the pіramidi:

18.5. Obsyag til wrapping.

The curve is visible, given equal to y = f (x). It is admitted that the function f (x) is non-discontinuous. As I draw the curvilinear trapezium with the bases a and b wrap around the axis tilo wrapping.

y = f (x)

Oskіlki dermal peretin tila area x = const є colo radius
Then the obsyag tila wrapping can easily be found behind the otriman vische formula:

18.6. The area of ​​the surface is wrapped.

M i B

value: Flat surface wrapping Crooked AB near the given axis call the boundary, until the area of ​​the surface of the wrap of lamanichs, inscribed in the curve AB, is pushed down to zero, the most common zines of lamanichs.

Rise an arc AB into n parts by points M 0, M 1, M 2, ..., M n. The coordinates of the vertices are from the rimano lamano, the coordinates x i і y i. When lamina is wrapped around the axis, it is possible to put on the surface, which can be folded from the side surfaces of the truncated cones, the area of ​​which is the road P i. The qia of the area can be known for the formula:

Here S i is the skin jordi.

Zastosov's Lagrange theorem (div. Lagrange's theorem) Before the announcement
.

1. The area of ​​the flat figuri.

The area of ​​the curved trapezium, surrounded by a non-negative function f (x), Vissy abscis and straight x = a, x = b, Start yak S = ∫ a b f x d x.

The area of ​​the crooked trapeze

Figuri area, interconnected by function f (x),, Start by the formula S = Σ i: f x ≥ 0 ∫ x i - 1 x i f x d x - Σ i: f x< 0 ∫ x i - 1 x i | f x | d x , где x i- zero functions. In other words, it is necessary to count the area of ​​the center of figurines, it is necessary to break out function zeros f (x) in part, integrate the function f on the skin of the viyshov of the prominence of the constancy of the sign, the area around the edges of the integral in the direction, on some functions f receive signs, and recognize from the first friend.

2. The area of ​​the crooked sector.

The area of ​​the crooked sector ρ = ρ (φ) in polar coordinate systems, de ρ (φ) - without interruption and non-negative on [α; β] function. Figura, surrounded by a curve ρ (φ) і exchanges φ = α , φ = β , To be called a curved sector. The area of ​​the curved sector of the road S = 1 2 ∫ α β ρ 2 φ d φ.

3. Obsyag tila wrapping.

Obsyag tila wrapping

Let it be wrapped around the axis OX curved trapezium, interlaced without interruption in the shape function f (x)... Yogo obsyag turn the formula V = π ∫ a b f 2 x d x.

Before the tasks about the knowledge of the volume of the body behind the area of ​​the transverse overrun

Nehay tilo is laid between areas x = aі x = b, And the area is cut by the area, so pass through the point x, - without interruption to the function σ (x)... Todi yogo obsyag road V = ∫ a b σ x d x.

4. Dovzhina arc crooked.

Do not give a curve r → t = x t, y t, z t t = αі t = β turn by the formula S = ∫ α β x 't 2 + y' t 2 + z 't 2 dt.

Dovzhina arc of a flat crooked Zokrem, dovzhina flat crooked, how to set on the coordinate area OXY rivnyannyam y = f (x), a ≤ x ≤ b, Swing by the formula S = ∫ a b 1 + f 'x 2 dx.

5. The area of ​​the surface of the wrapping.

The area of ​​the surface of the wrap Let the surface of the wrap be set on the axis OX graph of the function y = f (x), a ≤ x ≤ b, I function f I will go without interruption for a whole series of messages. The actual area of ​​the surface is wrapped by the formula Π = 2 π ∫ a b f x 1 + f 'x 2 d x.