The volume of the body covered by the wrappings of the cycloid arch. How to calculate the volume of body wrapping for the help of the singing integral? Calculation of the area of ​​the figure surrounded by lines, given parametrically

Lectures 8. Programs of the sing integral.

The addendum of the integral to the physical problems is based on the power of the additivity of the integral for the impersonal. Therefore, for the help of the integral, such quantities can be counted, as if they themselves are additive in multiplicity. For example, the area of ​​the figure is equal to the sum of the areas of the Dovzhin arc, the area of ​​the surface, the volume of the body, the mass of the body may have the same power. That number of quantities can be calculated with the help of a simple integral.

You can twist two methods and solve problems: method of integral sums and method of differentials.

The method of integral sums repeats the construction of a single integral: there will be splits, points are calculated, for which the function is calculated, the integral sum is calculated, the boundary transition is rotated. For whom, all the methods are basic folding - to bring what is between you and the same those that are necessary for the task.

Method of differentials victorist non-negligence integral the Newton-Leibnitz formula. Calculate the differential of the magnitude, as required, that buv, integrating the differential, following the Newton-Leibniz formula, take the required magnitude. Whom has the whole method of basic consistency - to bring what the differential of the required value has been calculated, and nothing else.

Calculation of the area of ​​flat figures.

1. The figure is surrounded by a graph of a function, given a Cartesian coordinate system.

We have come to understand the sing integral in terms of the area of ​​the curvilinear trapezoid (actually, the method of integral sums). This function only accepts see the meaning then the area under the graph of the function on the vіdrіzka can be calculated for the help of the sing integral. We respect that This is why the differential method can be used here.

But the function can also take negative values ​​on the other side, but the integral of the other side gives a negative area, which superimposes the designated area.

You can calculate the area using the formulaS=. It is essential to change the sign of the function in quiet areas, in which there are negative values.

If you need to calculate the area of ​​​​the figure, surrounded by the graph of the function to the beast, and below by the graph of the function, then you can use the formulaS= , so yak.

butt. Calculate the area of ​​the figure, surrounded by lines x=0, x=2 and graphs of functions y=x 2 , y=x 3 .

It is worth noting that the interval (0,1) has an unevenness x 2 > x 3 , and for x >1 an unevenness x 3 > x 2 . Tom

2. The figure is surrounded by a graph of the function, given in the system of polar coordinates.

Let the task function graph for the polar coordinate system and we want to calculate the area of ​​the curvilinear sector, surrounded by two, by changing the function graph for the polar coordinate system.

Here you can use the method of integral sums, calculating the area of ​​the curvilinear sector as between the sum of the areas of elementary sectors, in which the function graph is replaced by an arc of the stake .

You can twist the differential method: .

You can mirkuvati like this. Replacing the elementary curvilinear sector, which gives the central kut a circular sector, maybe a proportion. Zvіdsi . Integrating the Vicorist formula of Newton - Leibnitz, of course .

butt. Calculate the area of ​​​​the stake (perevirim formula). Dear. The area of ​​the stake is more expensive .

butt. I’m counting the area, I’m surrounded by cardio .

3 The figure is surrounded by a graph of the function specified by the parameters.

The function can be set parametrically as . Vikoristovuemo formula S= , substituting for her interintegration for a new change. . When you calculate the integral, you see those areas, the de-integral function may have the first sign and protect the entire area with this other sign.

butt. Calculate the area, surround it with an elіps.

Vikoristovuemo symmetry of the ellipse, counting the area of ​​the quarter of the ellipse, which is in the first quadrant. Whose quadrant? Tom.

Calculation of contacts tel.

1. Calculation of obsyagіv tіl for the areas of parallel reperіziv.

Let it be necessary to calculate the volume of the actual body V for the given areas of the cross section of the body with planes perpendicular to the straight line OX, passing through the point x of the straight line OX.

We need the method of differentials. Importantly, elementary volume, above the vertical volume of a straight circular cylinder with a base area and a height, is taken . Integrating and zastosovuyuchi Newton-Leibniz formula, we take

2. The calculation is obsyagіv tіl wrapping.

Let it be necessary to virahuvati OX.

Todi .

Similarly, volumeOY If a function is given to a viewer, it can be calculated using a formula.

This function is set for the viewer and it is necessary to determine the volume of the body wrap around the axisOY the formula for the calculation of the obligation can be taken off by the coming rank.

Passing to the differential and not using quadratic terms, perhaps . Integrating and zastosovuyuchi Newton-Leibniz formula, maybe.

butt. Calculate obsyag cooli.

butt. Calculate the volume of a right circular cone surrounded by a surface area.

Let's calculate the volume, like the volume of the body of the wrapper, made around the OZ axis of the straight-cut tricot in the OXZ plane, the leg of which lies on the OZ axis and is straight z = H, and the hypotenuse lies on the straight line.

Turning x through z, we can take .

Calculate the length of the arc.

In order to take the formulas for calculating the back of the arc, we have created in 1 semester the formula for the differential of the back of the arc.

Like an arc in a graph of an uninterruptedly differentiated function, the differential of the second arc can be calculated using the formula

. Tom

Even though a smooth arc is given parametrically, then

. Tom .

Likewise, the arc is set in the polar coordinate system, then

. Tom .

butt. Unravel the edge of the arc of the graph of the function, . .

First, go to the formulas for the area of ​​the surface wrap, for a brief formula of the surface wrap itself. The top wrapping, or, what are the same - the top body wrapping - a spacious figure, the wrapping is made into a vіdrіzka AB curve on the axis Ox(Figure below).

I will reveal the curvilinear trapeze, I will surround the beast with the guessing curve of the curve. Tіlo, made for wrappings tsієї trapezії navko tiєї zh osі Ox and є tіlo wrapping. And the area of ​​the surface wrapping or the surface of the body wrapping is the whole yogo ovnishnya shell, not rahuyuchi kіl, utavleny wrappings on the axis straight x = aі x = b .

Respectfully, that the body of the wrapping and, obviously, the same surface can be made so that the wrappings of the figure are not on the axis. Ox, but about axis Ouch.

Calculation of the surface area of ​​the wrapping, given in rectangular coordinates

Let's go at rectangular coordinates on the flat plane y = f(x) a curve is given, wrapping around the coordinate axis is given a wrapping body.

The formula for calculating the surface area of ​​the wrap:

(1).

example 1. Know the area of ​​the surface of the paraboloid covered by wraps around the axis Ox parabolic arcs that change x view x= 0 to x = a .

Solution. We can clearly see the function, as we set the arc of the parabola:

We know the following functions:

First, let's speed up the formula for knowing the area of ​​the surface wrapping, let's write that part of the її pіdіntegralny virase, as a root and conceivably there is only known pokhіdn:

Vidpovіd: dozhina arc crooked dorіvnyuє

.

butt 2. Know the area of ​​the surface that wraps around the axis Ox astroidi.

Solution. It is enough to calculate the area of ​​the surface, which will go into the wrapper of one astroid needle, ruffled in the first quarter, and multiply її by 2. From the alignment of the astroid, it is clearly a function, so we will need to introduce a formula for calculating the surface area:

.

Variable integration from 0 to a:

Calculation of the surface area of ​​the wrapping, given parametrically

We can look at the slope, if the curve that sets the surface of the wrap is set by parametric equalities

The same area of ​​surface wrapping is calculated according to the formula

(2).

example 3. Know the area of ​​the surface wrapping, covered with wrappings on the axis Ouch figure, surrounded by a cycloid and a straight line y = a. The cycloid is given by parametric equalities

Solution. We know the crossing point of a cycloid and a straight line. Aligning the alignment of cycloids and alignment of straight lines y = a, we know

Why do you see what interintegration is showing

Now we can fill in formula (2). Let's know the fun:

We write down the root of the virase in the formula, representing the known results:

We know the root of this virus:

.

Suppose we have found the formula (2):

.

Let's make a substitution:

I, nareshti, we know

The converted viruses have different trigonometric formulas

Suggestion: the area of ​​the surface wrapping is good.

Calculation of the surface area of ​​the wrapping, given in polar coordinates

Let the curve wrap around the surface, set in polar coordinates.

Just as for the purpose of knowing the area, the need for newcomers to wake up an armchair is not the most important (shards integrated by the power forces will often be light). It is possible to master the correct swidka technique and encourage graphs with the help of methodical materials and geometric transformations of graphs. Ale, vlasne, about the importance of the armchair, I have already spoken more than once in the lesson.

In the integral calculation, there are a lot of other additions, for the help of the simple integral, you can calculate the area of ​​the figure, the volume of the body of the wrap, the length of the arc, the area of ​​the surface wrap and the rich other. Tom will have fun, be kind, tune in an optimistic way!

Give deak a plane figure on the coordinate plane. Have you noticed? ... Tsikavo, who imagined what ... =))) We already knew. Ale, moreover, you can twist the whole figure, and twist it in two ways:

- about the abscissa axis;
- About the axis of ordinates.

In this article, insults will be sorted out. Especially, there is another way of wrapping, which leads to the most difficulties, but in fact the solution is practically the same, as in the extended wrapping around the abscissa axis. Yak bonus I'll turn up tasks, and we will tell you, as if you know the area in a different way - along the axis. Navit not so bonus, but the material successfully fits into the topic.

Probably the most popular wrapping variety.

flat figures on the axis

butt 1

Calculate the volume of the body, taken off the wrapping of the figure, surrounded by lines around the axis.

Solution: Yak i in the task of rebuking the area, solutions start from the armchair of a flat figure. Tobto, on the square it is necessary to induce a figure, surrounded by lines, with which it is not forgotten that it is equal to set everything. As a rational and smarter vikonati armchair, you can recognize on the sides Graphs and Powers of Elementary Functionsі The value of the integral. How to calculate the area of ​​\u200b\u200bthe figure. Tse Chinese nagaduvannya, and at the same time I no longer sing.

Armchair here to finish the forgiveness:

As a result, the wrapper will come out such an egg-shaped flying plate, as symmetrical as it is to the axis. In truth, the body has a mathematical name, but after the finish, it was possible to clarify the molt, they gave it a demo.

How to designate body wrapping?

The volume of body wrapping can be calculated using the formula:

The formula has a number before the integral. So it happened - everything that is spinning in life is tied up with a cієyu constant.

How to set up the interintegration between “a” and “be”, I guess, it’s easy to guess from the vikon’s armchair.

What is the function? Let's look at the chair. The flat figure is surrounded by a graph of a parabola to the beast. This is the function that is at the mercy of the formula.

In practical tasks, a flat figure can sometimes be roztashovuvatisya lower than the axis. We don’t change anything - the integrand function of the formula is squared: in this order integral is always unknown which is logical.

Let's calculate the volume of body wrapping, vicorist formula:

As I have already appointed, the integral can always come out, let's forgive, smut, but respect.

Vidpovid:

In the case of the species, it is necessary to indicate the difference in terms of volume - cubic units. So our tili has a wrapper of approximately 3.35 "cubes". Why is cubic alone? Because the most universal formula. They can be cubic centimeters, they can be cubic meters, they can be cubic kilometers, etc.

butt 2

Know the volume of the body, wrapped around the axis of the figure, surrounded by lines,

This is an example of an independent solution. Outwardly, the solution is that it is similar to the lesson.

Let's take a look at two folding tasks, which are often used in practice.

butt 3

Calculate the volume of the body, taken off when wrapped around the abscissa axis of the figure, surrounded by lines,

Solution: It is depicted on an armchair flat figure, surrounded by lines

Shukan's figure is shaded with blue color. With її wrapping on the axis, such a surreal bagel comes out of chotirma kutami.

The volume of body wrapping is calculable retail sales.

On the back of the head, a figure is seen, circled with a red color. When її wrapped around the axis, a truncated cone comes out. Significantly contracting the cut cone through .

Let's look at the figure, as circled green color. If you wrap the figure around the axis, then you will also see a truncated cone, only a little less. Significantly yogo obsyag through.

І, obviously, the price was tight - it was exactly the cut of our "donut".

We use the standard formula for determining the volume of body wrapping:

1) The figure, circled with a red color, is surrounded by a straight beast, to that:

2) The figure, circled in green color, is surrounded by a straight beast, to that:

3) Obsyag of a shukany body wrap:

Vidpovid:

Tsikavo, what's in to this particular type the solution can be reversed, vikoristovuyuuchi school formula for calculating the obligation of a clear cone.

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

Now we have a little trouble, and let's talk about geometric illusions.

People are often blamed for illusions, tied with obligations, as if remembering Perelman (the next) at the book Tsikava geometry. To marvel at the flat figure of the tall leader - it’s not large beyond the square, and the volume of the body wrapping becomes more than 50 cubic units, which is too big. To the point, an average statistical person for all his life sips his homeland with a volume of 18 square meters from the room, which, on the other hand, looks like a small volume.

Vzagali, the lighting system in the SRSR was really the best. That very book of Perelman, seen in 1950, is more good at developing, as the humorist said, mirkuvannya and read jokes and original non-standard solutions to problems. Recently, with great interest, having re-read the deacons, I recommend that they be accessible to the humanities. No, you don’t need to laugh, that I uttered an hour spent without a show, erudition, that wide horizons of a colloquial - a miraculous thing.

After a lyrical approach, I will say the words to the creative task:

butt 4

Calculate the volume of the body, wrapped around the axis of the flat figure, surrounded by lines, , de.

This is an example of an independent solution. To give respect that all the blues are in the smoothies, that is, in fact, given the inter-integration is ready. Correctly assign the graphs of trigonometric functions, guess the material for the lesson about geometric transformations of graphics: so the argument is divided into two: , then the graphs are stretched along the axis of increase. Bazhano know hocha b 3-4 specks for trigonometric tables, to be more precise vikonati kreslennya. Outwardly, the solution is that it is similar to the lesson. Before the speech, the task can be done rationally and not rationally.

Calculation of the volume of the body covered with wrappers
flat figures on the axis

Another paragraph will be tsіkavіshim, lower first. Task for calculating the volume of the body wrapping around the axis of ordinates - also to complete part of the guest control robots. I would like to take a look zavdannya about the significance of the figure area In another way - integrating along the axis, which will allow you not only to improve your skills, but also to learn to know the best way to solve. Tshomu has a practical life sense! As if out of a smile, my teacher was guessing at the methodology of teaching mathematics, a lot of graduates said: “Your subject has already helped us, now we are efficient managers and optimally cared for by staff.” With a bark of reward, I also hang my great tribute, even more so, that the knowledge of vicorist is taken away for direct confession =).

I recommend for reading to all new teapots. Moreover, the assimilation of the material of another paragraph will provide an invaluable help in calculating the underlying integrals.

butt 5

A flat figure is given, surrounded by lines , , .

1) Find the area of ​​a flat figure, surrounded by given lines.
2) Know the volume of the body, taken from the wrappings of a flat figure, surrounded by these lines along the axis.

Respect! Navite if you want to know only about another point, back to back obov'azkovo read first!

Solution: The order is made up of two parts. Mostly from the area.

1) Vikonaemo armchairs:

It is easy to remember that the function defines the upper arm of the parabola, and the function determines the lower arm of the parabola. Before us is a trivial parabola, like lying on the boots.

A figure is needed, the area that I need to know is shaded with a blue color.

How to know the area of ​​\u200b\u200bthe figure? You can know in a “magnificent” way, looking at the lessons The value of the integral. How to calculate the area of ​​\u200b\u200bthe figure. Moreover, the area of ​​\u200b\u200bthe figure is known as the sum of the area:
- On the vіdrіzku ;
- On the vіdrіzku.

Tom:

Why do different filth have a simple path of solution? First, there were two integrals. In another way, the root in integrals, and the root in integrals is not a gift, before that you can get lost in the process of integration. Really, they integrated, obviously, not in a big way, but in practice everything is significantly summed up, I just chose the “better” function for setting the function.

Є more rational way of solution: vіn polygaє in the transition to turning functions and integration along the axis.

How to go to the return functions? Roughly seeming, it is necessary to say "iks" through "igrok". Let's take a guess at the parabola:

What is sufficient, but we can change it, that such a function itself can be introduced from the bottom nail:

From a straight line, everything is simpler:

Now we marvel at the sky: be kind, periodically heal your head to the right by 90 degrees along the explanation (not a joke!). It is necessary for us to lie down on the vіdrіzku, which is the meaning of a red dotted line. At the same time, the straight line is ruffled more than a parabola, and therefore, the square of the figure should be known for the formula already known to you: . What has changed in the formula? Only letters, and no more.

! Note: Between the integration along the axis of the next line strictly downhill!

We know the area:

On a vіdrіzku, to that:

To restore respect, as I have achieved integration, a rational way, and at the offensive point of the task, I will understand why.

For readers who doubt the correctness of integration, I know better:

We removed the integrand function, and then the integration was correct.

Vidpovid:

2) Let's calculate the volume of the body, put on the wrappings of the circle of the figure, about the axis.

I will repaint the chairs of the troch in a different design:

Otzhe, the figure, shaded with blue color, wraps around the axle. As a result, a “high blizzard” emerges, which spins around its own axis.

To understand the volume of the body, the wrapper is integrable along the axis. From the beginning it is necessary to go to the turning functions. It is already broken and painted in detail at the front point.

Now I’m shaking my head again to the right and we’re twisting our figure. It is obvious that the volume of body wrapping should be known as the volume difference.

It wraps a figure, circled with a red color, on the axis, as a result, a vision cone emerges. Significantly tsey obsyag through.

Rotate the figure, circled in green color, on the axis it is signified through the volume of the body of the wrapped body.

Obsjag of our blizzard is expensive retail obsyagiv.

Vikoristovuёmo formula for knowing the volume of body wrapping:

Why do you see the formula of the front paragraph? Less than a letter.

And the axis and the progress of integration, about the yak, I recently spoke, it’s easier to know lower in front lead to the integrand function in the 4th step.

Vidpovid:

However, a sickly blizzard.

To turn respect, if you want to wrap a flat figure around the axis, then we need to wrap it differently, naturally, about it.

butt 6

A flat figure is given, surrounded by lines, but it is vissyu.

1) Go to the return functions and find the area of ​​a flat figure, surrounded by given lines, integrating over changes.
2) Calculate the volume of the body, taken off the wrappings of the flat figure, surrounded by these lines along the axis.

This is an example of an independent solution. Bajayuchi can also know the area of ​​\u200b\u200bthe figure in a “superior” way, having done this reversal of paragraph 1). And from what, I repeat, you wrap a flat figure on the axis, then we’ll see more body wrapping with a smaller obliga, to speech, the correct statement (tezh for lovers of virishuvati).

Outwardly, the decision was made on two points of order for the task of the lesson.

So, do not forget to heal your head with a right-handed person, so that you can grow up in the bodies of the wrapper and in the boundaries of integration!

I love you, students of VNZ Argemony!

More trohi - and the course will be completed, and at once we will take care of the axis.

Zhouli trohi waved her hand - and in the wind appeared to stand. Or rather, it was a rectilinear trapezium. Vaughn just hung in the air, created by magical energy, as it flowed along її sides, and also swirled in the middle of the trapezium itself, through which it all vibrated and shimmered.
Then the vikladach trohi pomited a circular motion with the fingers of his hand - and the trapezium began to wrap itself around an invisible axis. Quite rightly, then we’ll get better and better - so, that in the future, the volume of posting began to appear. It seemed that the magical energy rose from her.

Dali trapilos like this: the gleaming contours of the figure and її insides began to become like speech, the light became less and less memorable, then the figure itself became more and more similar to a strange one. The grains of the material were gradually divided according to the figure. The first axis was all gone: the wrapping, the candle. Povitri visiv has an object similar to a virva. Zhouly carefully slipped the yogo onto the table.

Well axis. Approximately in this way it is possible to materialize a lot of objects - with a wrapping way, like flat figures that are almost straight lines. Obviously, for materialization, a small amount of speech is needed, so as to fill with oneself the entire volume, which is settled and temporally subdued for additional magical energy. And the axis, in order to exactly cheer up, how much speech is required, - it is necessary to know the body that is accepted. Otherwise, if there is not enough speech, it will not be possible to cover the whole volume with oneself and the body can be German, with vadas. And the materials are even more embellished with the great superfluity of speech - it’s not necessary to exude magical energy.
Well, how is it that we have a lot of speech? Todi, in addition to counting obsyagi tel, you can estimate, as for rozmirami tіlo we can grow without special amounts of magical energy.
Any excess of the received material is another thought. Where will the superfluous speeches go? Obsipayutsya, being not zadіyanimi? Chi stick on the body of abyak?
There's more to think about here. As soon as you had some thoughts, then I listened to them out of satisfaction. In the meantime, let's move on to the calculation of obsyagiv tіl, taking away such a way.
Here one can see a sprat of vipadkiv.

Vipadok 1.

The area, as we will wrap, is the most classic curvilinear trapezium.

It is obvious that we can only wrap around the axis OH. How can I destroy the right-handed trapezoid horizontally so that it doesn’t overwhelm the whole OY, you can wrap it around and around the axis. Spell formulas for both vipadkіv are as follows:

We are with you already do good mastered the basics magic splash on the function, I think it’s not difficult for you to transfer the figure so in the coordinate axes, so that it’s handy for working with it.

Vipadok 2

You can wrap not only the classic curvilinear trapezium, but the figure of such a look:

When wrapping, we take away our own ring. And having transferred the figure to the positive area, we can її wrap and choose the axis OY. Tezh otrimaєmo kіltse chi nі. To lay everything in the way that the figure is roztashovuvatym: if you pass the boundary exactly along the axis OY, then the ring will not be seen. It is possible to unravel the obsyagi of such tіl wrappings, using the following incantation:

Vipadok 3.

Let's guess, we have wonderful curves, but such that they are asked not in a way that is familiar to us, but in a parametric way. Such curves are often closed. The parameter t is guilty of changing in such a way that the closed figure, when bypassing її with a curve (cordon), would become angry.

Then for the calculation of the volume of the body, the wrapping should be done on the axis OH or OY, you need to cast such a spell:

Qi formulas can be twisted in the direction of non-closed curves: if obedience ends lie on the axis OX and the axis OY. The figure appears closed in any way: the ends close the axis.

Vipadok 4.

Some of the magic curves are given by polar coordinates (r=r(fi)). You can wrap the same figure around the polar axis. In this direction, the Cartesian coordinate system goes down from the polar and lies
x = r (fi) * cos (fi)
y=r(fi)*sin(fi)
Thus, we come to the parametric look of the curve, where the parameter fi is obliged to change so that when going around the curve, the area becomes left-handed.
І koristuєmosya incantational formulas z nagodi 3.

However, for vipadku polar coordinates є і has its own incantation formula:

Obviously, flat figures can be wrapped around as well as any other straight lines, not just the axes OX and OY, but if manipulations are already folded, we will be surrounded by those twists, which were discussed in the lecture.

And now homework . I am not giving you specific figures. We have already developed a lot of functions, and I want you to design it yourself in such a way that you might need it in magical practice. I think that there will be enough examples for all indications at the lecture.