Graph limited online lines. Calculate the area of \u200b\u200bthe shape, limited lines

Certain integral. How to calculate the area of \u200b\u200bthe figure

Go to consideration of integral application applications. In this lesson, we will analyze the typical and most common task. - How to calculate the plane shape with a specific integral. Finally, seeing meaning in higher mathematics - will find it. Little. We'll have to bring closer in life country cottage area Elementary functions and find its area using a specific integral.

For successful material development, it is necessary:

1) understand uncertain integral at least at the average level. Thus, teapotes should be familiar with the lesson Not.

2) To be able to apply the Newton labnic formula and calculate a specific integral. To establish warm friendships with certain integrals on the page Certain integral. Examples of solutions.

In fact, in order to find the area of \u200b\u200bthe figure, there is no such knowledge of the uncertain and defined integral. The task "Calculate the area with the help of a specific integral" always implies the construction of the drawing, so much more actual question There will be your knowledge and skills to build drawings. In this regard, it is useful to refresh in the memory of graphics of the main elementary functions, and at least, be able to build a straight, parabola and hyperbola. It can be done (many - needed) with methodical material and articles on geometric transformations of graphs.

Actually, with the task of finding an area with the help of a specific integral, everyone is familiar from school, and we will eat little forward from school program. This article could not even be, but the fact is that the task is found in 99 cases out of 100, when the student suffers from a hateful tower with enthusiasm departing the course of higher mathematics.

Materials of this workshop are presented simply, in detail and with a minimum of theory.

Let's start with a curvilinear trapezium.

Curvilinear trapezium A flat figure is called a limited axis, straight, and a continuous schedule on a segment of a function that does not change the sign on this interval. Let this figure be located not less The abscissa axis:

Then the area of \u200b\u200bthe curvilinear trapezium is numerically equal to a specific integral. Any particular integral (which exists) has a very good geometric meaning. At the lesson Certain integral. Examples of solutions I said that a certain integral is a number. And now it's time to state one more useful fact. From the point of view of geometry, a certain integral is an area.

I.e, a specific integral (if it exists) geometrically corresponds to the area of \u200b\u200bsome figure. For example, consider a specific integral. The integrand function sets a curve on the plane, located above the axis (which wishes can draw the drawing), and the specific integral itself is numerically equal to the area of \u200b\u200bthe corresponding curvilinear trapezium.

Example 1.

This is a typical task formulation. First I. the most important thing Solutions - Building drawing. And the drawing must be built RIGHT.

When building a drawing, I recommend the following order: first it is better to build all straight (if they are) and only later - Parabolas, hyperbolas, schedules of other functions. Function graphs are more profitable to build potochoe, with the technique of check-in construction can be found in reference material Charts and properties of elementary functions. There you can also find a very useful material in relation to our lesson the material - how to quickly build a parabola.

In this task, the decision may look like that.
Perform the drawing (note that the equation sets the axis):

I will not stroke a curvilinear trapeze, here it is obvious about which area this is speech. The decision continues like this:

On the segment schedule a function is located over the axis, so:

Answer:

Who has difficulties with the calculation of a certain integral and the use of Newton-Leibnia formula , refer to the lecture Certain integral. Examples of solutions.

After the task is completed, it is always useful to look at the drawing and estimate, the real one turned out. In this case, "on the eyes" we count the number of cells in the drawing - well, approximately 9 will be flown, it seems to the truth. It is quite clear that if we had, say, answer: 20 square units, it is obvious that an error is made somewhere - in the figure of 20 cells, it is clearly not fitted, from the strength of a dozen. If the answer turned out negative, the task is also decided incorrectly.

Example 2.

Calculate the area of \u200b\u200bthe figure, limited lines , and axis

This is an example for self-decide. Complete solution And the answer at the end of the lesson.

What to do if the curvilinear trapezium is located under the axis?

Example 3.

Calculate the area of \u200b\u200bthe shape, limited lines, and the coordinate axes.

Decision: Perform drawing:

If the curvilinear trapezium is located under the axis (or at least not higher This axis), then its area can be found by the formula:
In this case:

Attention! Do not confuse two types of tasks:

1) If you are invited to solve a simple integral without any geometric meaning, then it may be negative.

2) If you are invited to find the figure of the figure using a specific integral, then the area is always positive! That is why in just the considered formula appears minus.

In practice, the figure is most often located in the upper and lower half plane, and therefore, from the simplest school charts, go to more meaningful examples.

Example 4.

Find the area of \u200b\u200ba flat figure, limited lines ,.

Decision: First you need to draw a drawing. Generally speaking, when building a drawing in tasks to the area, we are most interested in the intersection points of the lines. Find points of intersection of parabola and direct. This can be done in two ways. The first method is analytical. We solve the equation:

So, the lower integration limit, the upper limit of integration.
This way is better, if possible, do not use.

It is much more profitable and faster to build the lines of the line, while the integration limits are clarified as if "by themselves". The technique of the cessation for various graphs is considered in detail in the help Charts and properties of elementary functions . However, an analytical way to find the limits after all, it is sometimes necessary to apply if, for example, the schedule is large enough, or a trained construction did not reveal the integration limits (they can be fractional or irrational). And such an example, we also consider.

We return to our task: more rational first build a straight line and only then Parabola. Perform drawing:

I repeat that in the current construction, the integration limits are most often found out by the "automatic".

And now the working formula: If on the segment some continuous function more or equal Some continuous function, the area of \u200b\u200bthe figure, limited by graphs of these functions and direct, can be found by the formula:

Here it is no longer necessary to think where the figure is located - over the axis or under the axis, and, roughly speaking, important what is the graph above(relative to another schedule) and what - below.

In this example, it is obvious that on the segment of Parabola is located above straight, and therefore it is necessary to subtract

Completion of the solution may look like this:

The desired figure is limited to parabola from above and direct bottom.
On the segment, according to the corresponding formula:

Answer:

In fact, the school formula for the area of \u200b\u200bthe curvilinear trapezium in the lower half-plane (see simple example No. 3) - private case Formulas . Since the axis is defined by the equation, and the function graph is located not higher Axis, T.

And now a couple of examples for an independent decision

Example 5.

Example 6.

Find the area of \u200b\u200bthe figure limited lines ,.

In the course of solving tasks for calculating the area with a specific integral, a funny case occurs sometimes. The drawing is completed correctly, calculations - right, but intensified ... found the area is not the figureThat this is how your humble servant was packed. Here is a real case from life:

Example 7.

Calculate the area of \u200b\u200bthe shape, limited lines ,,,.

Decision: First do the drawing:

... oh, the drawing of Khrenovynsky came out, but everything seems to be picking up.

Figure whose area we need to find is shaded in blue (Look carefully on the condition - than the figure is limited!). But in practice, "glitch" often arises in mindfulness, which you need to find an area of \u200b\u200bthe figure, which is shaded green!

This example is still useful and the fact that in it the area of \u200b\u200bthe figure is considered using two specific integrals. Really:

1) A straight schedule is located on the segment over the axis;

2) On the segment over the axis there is a graph of hyperboles.

It is clear that the square can (and need) to decompose, so:

Answer:

Go to another substantive task.

Example 8.

Calculate the area of \u200b\u200bthe shape, limited lines,
Imagine the equation in the "school" form, and perform the current drawing:

From the drawing it is clear that the upper limit we have "good" :.
But what is the lower limit?! It is clear that this is not an integer, but what? May be ? But where is the guarantee that the drawing is made with perfect accuracy, it may well be that. Or root. And if we generally improperly built a schedule?

In such cases, have to spend additional time and specify the integration limits analytically.

Find the intersection points of the direct and parabola.
To do this, solve the equation:

,

Indeed.

Further solution is trivial, the main thing is not to get confused in substitutions and signs, the calculations here are not the simplest.

On cut According to the corresponding formula:

Answer:

Well, and in the conclusion of the lesson, consider two tasks more difficult.

Example 9.

Calculate the area of \u200b\u200bthe shape, limited lines ,,

Decision: Show this shape in the drawing.

Damn, forgot the schedule to sign, but to redo the picture, sorry, not a hotz. Not inherited, shorter, day today \u003d)

For check-in construction you need to know appearance sinusoids (and it is generally helpful to know graphs of all elementary functions), as well as some sinus values, they can be found in trigonometric table. In some cases (as in this), it is allowed to build a schematic drawing on which the graphs and integration limits must be reflected in principle.

With the limits of integration, there are no problems here, they follow directly from the condition: - "X" varies from zero to "pi". We draw up a further solution:

On the segment, the function graph is located above the axis, so:

In fact, in order to find the area of \u200b\u200bthe figure, there is no such knowledge of the uncertain and defined integral. The task "Calculate the area with the help of a specific integral" always implies the construction of the drawingTherefore, a much more relevant issue will be your knowledge and skills of building drawings. In this regard, it is useful to refresh in the memory of graphics of basic elementary functions, and at least be able to build a straight, and hyperbola.

The curvilinear trapezion is called a flat figure, limited to the axis, straight, and a continuous schedule on a segment of a function that does not change the sign on this interval. Let this figure be located not less The abscissa axis:

Then the area of \u200b\u200bthe curvilinear trapezium is numerically equal to a specific integral. Any particular integral (which exists) has a very good geometric meaning.

From the point of view of geometry, a certain integral is an area.

I.e, A specific integral (if it exists) geometrically corresponds to the area of \u200b\u200bsome shape. For example, consider a specific integral. The integrand function sets a curve on the plane, located above the axis (which wishes can draw the drawing), and the specific integral itself is numerically equal to the area of \u200b\u200bthe corresponding curvilinear trapezium.

Example 1.

This is a typical task formulation. The first and most important point of the decision - building a drawing. And the drawing must be built RIGHT.

In this task, the decision may look like that.
Perform the drawing (note that the equation sets the axis):

On the segment schedule a function is located over the axis, so:

Answer:

Example 3.

Calculate the area of \u200b\u200bthe shape, limited lines, and the coordinate axes.

Decision: Perform drawing:

If the curvilinear trapezium is located under the axis(or at least not higher This axis), then its area can be found by the formula:

In this case:

Attention! Do not confuse two types of tasks:

1) If you are invited to solve a simple integral without any geometric meaning, then it may be negative.

2) If you are invited to find the figure of the figure using a specific integral, then the area is always positive! That is why in just the considered formula appears minus.

In practice, the figure is most often located in the upper and lower half plane, and therefore, from the simplest school charts, go to more meaningful examples.

Example 4.

Find the area of \u200b\u200ba flat figure, limited lines ,.

So, the lower integration limit, the upper limit of integration.

This way is better, if possible, do not use.

It is much more profitable and faster to build the lines of the line, while the integration limits are clarified as if "by themselves". However, an analytical way to find the limits after all, it is sometimes necessary to apply if, for example, the schedule is large enough, or a trained construction did not reveal the integration limits (they can be fractional or irrational). And such an example, we also consider.

We return to our task: more rational first build a straight line and only then Parabola. Perform drawing:

In this example, it is obvious that on the segment of Parabola is located above straight, and therefore it is necessary to subtract

Completion of the solution may look like this:

The desired figure is limited to parabola from above and direct bottom.
On the segment, according to the corresponding formula:

Answer:

Example 4.

Calculate the area of \u200b\u200bthe shape, limited lines ,,,.

Decision: First do the drawing:

This example is still useful and the fact that in it the area of \u200b\u200bthe figure is considered using two specific integrals.

Really:

1) A straight schedule is located on the segment over the axis;

2) On the segment over the axis there is a graph of hyperboles.

It is clear that the square can (and need) to decompose, so:

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The iterative algorithm for constructing the menger sponge is quite simple: the source cube with a side 1 is divided by planes parallel to its faces, on 27 equal cubes. One central cube and 6 adjacent cubes are removed from it. A set consisting of 20 remaining smaller cubes is obtained. By doing the same with each of these cubes, we obtain a set, consisting already from 400 smaller cubes. Continuing this process infinitely, we get a sponge of Menger.

From this article, you will learn how to find an area of \u200b\u200bfigures limited by lines using calculations using integrals. For the first time, we encounter such a task in high school, when we just passed the study of certain integrals and it is time to begin the geometric interpretation of the knowledge gained in practice.

So, what will be required to successfully solve the problem of searching for an area of \u200b\u200bthe figure with the help of integrals:

Skill competently build drawings;
The ability to solve a specific integral with the help of the well-known Newton-Leibnic formula;
The ability to "see" a more profitable solution option - i.e. Understand how in such an case it will be more convenient to carry out integration? Along the X axis (ox) or the axis of the game (OY)?
Well, where without correct computing?) This includes an understanding how to solve that other type of integrals and the correct numerical calculations.

The algorithm for solving the task of calculating the area of \u200b\u200bthe figure, limited lines:

1. Build a drawing. It is advisable to do it on a piece in a cage, with a large scale. We subscribe a pencil over each chart the name of this function. The signature of the graphs is made exclusively for the convenience of further computing. After receiving a graph of the desired figure, in most cases it will be seen immediately which integration limits will be used. So we solve the task graphic method. However, it happens that the values \u200b\u200bof the limits are fractional or irrational. Therefore, you can make additional calculations, go to step two.

2. If the integration limits are clearly not specified, we find the intersection points of the graphs with each other, and we look at whether our graphic solution with analytical is coincided.

3. Next, it is necessary to analyze the drawing. Depending on how the graphics of functions are located, exist different approaches To finding the area of \u200b\u200bthe figure. Consider different examples To find the area of \u200b\u200bthe figure with the help of integrals.

3.1. The most classic and simple task option is when you need to find the area of \u200b\u200bthe curvilinear trapezium. What is a curvilinear trapeze? This is a flat figure limited to X axis (y \u003d 0)straight x \u003d a, x \u003d b and any curve continuous on the interval from a. before b.. At the same time, this figure is non-negative and is not lower than the abscissa axis. In this case, the area of \u200b\u200bthe curvilinear trapezium is numerically equal to a specific integral calculated by the Newton Labender formula:

Example 1. y \u003d x2 - 3x + 3, x \u003d 1, x \u003d 3, y \u003d 0.

What lines is the figure limited? We have parabola y \u003d x2 - 3x + 3which is located above the axis OH, it is non-negative, because all points of this parabola have positive meanings. Next, the direct x \u003d 1. and x \u003d 3.who run parallel to the axis OUare the restrictive lines of the figure on the left and right. Well y \u003d 0.She is an X axis that limits the figure below. The resulting figure is shaded, as can be seen from the drawing on the left. In this case, you can immediately start solving the problem. We have a simple example of a curvilinear trapezium, which is further solving with the help of Newton-Leibnic formula.

3.2. In the previous paragraph 3.1, the case is disassembled when the curvilinear trapezium is located above the X axis. Now consider the case when the conditions of the task are the same, except that the function runs under the X axis. The standard Newton-Labender formula is added minus. How to solve such a task Consider further.

Example 2. . Calculate the area of \u200b\u200bthe shape, limited lines y \u003d x2 + 6x + 2, x \u003d -4, x \u003d -1, y \u003d 0.

IN this example We have parabola y \u003d x2 + 6x + 2which originates from the axis OH, straight x \u003d -4, x \u003d -1, y \u003d 0. Here y \u003d 0. Limits the desired figure from above. Straight x \u003d -4. and x \u003d -1. These are borders within which a specific integral will be calculated. The principle of solving the problem of finding the area of \u200b\u200bthe figure almost completely coincides with the example number 1. The only difference is that set function not positive, and everything is also continuous on the interval [-4; -1] . What does not mean positive? As can be seen from the figure, the figure, which lies within the specified ICs, has exclusively "negative" coordinates, which we need to see and remember when solving the problem. The area of \u200b\u200bthe figure is looking for Newton Labitsa formula, only with a minus sign in the beginning.

The article is not completed.

So, what will be required to successfully solve the problem of searching for an area of \u200b\u200bthe figure with the help of integrals:

Skill competently build drawings;
The ability to solve a specific integral with the help of the well-known Newton-Leibnic formula;
The ability to "see" a more profitable solution option - i.e. Understand how in such an case it will be more convenient to carry out integration? Along the X axis (ox) or the axis of the game (OY)?
Well, where without correct computing?) This includes an understanding how to solve that other type of integrals and the correct numerical calculations.

The algorithm for solving the task of calculating the area of \u200b\u200bthe figure, limited lines:

1. Build a drawing. It is advisable to do it on a piece in a cage, with a large scale. We subscribe a pencil over each chart the name of this function. The signature of the graphs is made exclusively for the convenience of further computing. After receiving a graph of the desired figure, in most cases it will be seen immediately which integration limits will be used. Thus, we solve the task with a graphic method. However, it happens that the values \u200b\u200bof the limits are fractional or irrational. Therefore, you can make additional calculations, go to step two.

2. If the integration limits are clearly not specified, we find the intersection points of the graphs with each other, and we look at whether our graphic solution with analytical is coincided.

3. Next, it is necessary to analyze the drawing. Depending on how graphics of functions are located, there are different approaches to finding the area of \u200b\u200bthe figure. Consider different examples to find the area of \u200b\u200bthe figure with the help of integrals.

Example 1. y \u003d x2 - 3x + 3, x \u003d 1, x \u003d 3, y \u003d 0.

What lines is the figure limited? We have parabola y \u003d x2 - 3x + 3which is located above the axis OH, it is non-negative, because All points of this parabola are positive. Next, the direct x \u003d 1. and x \u003d 3.who run parallel to the axis OUare the restrictive lines of the figure on the left and right. Well y \u003d 0.She is an X axis that limits the figure below. The resulting figure is shaded, as can be seen from the drawing on the left. In this case, you can immediately start solving the problem. We have a simple example of a curvilinear trapezium, which is further solving with the help of Newton-Leibnic formula.

Example 2. . Calculate the area of \u200b\u200bthe shape, limited lines y \u003d x2 + 6x + 2, x \u003d -4, x \u003d -1, y \u003d 0.

In this example, we have a parabola y \u003d x2 + 6x + 2which originates from the axis OH, straight x \u003d -4, x \u003d -1, y \u003d 0. Here y \u003d 0. Limits the desired figure from above. Straight x \u003d -4. and x \u003d -1. These are borders within which a specific integral will be calculated. The principle of solving the problem of finding an area of \u200b\u200bthe figure almost completely coincides with the example number 1. The only difference is that the specified function is not positive, and everything is also continuous on the interval [-4; -1] . What does not mean positive? As can be seen from the figure, the figure, which lies within the specified ICs, has exclusively "negative" coordinates, which we need to see and remember when solving the problem. The area of \u200b\u200bthe figure is looking for Newton Labitsa formula, only with a minus sign in the beginning.

The article is not completed.

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