Definite integral. How to calculate the area of a figure
Let's move on to consider applications of integral calculus. In this lesson we will analyze the typical and most common task – how to use a definite integral to calculate the area of a plane figure. Finally, those who are looking for meaning in higher mathematics - may they find it. You never know. We'll have to bring it closer in life summer cottage plot elementary functions and find its area using a definite integral.
To successfully master the material, you must:
1) Understand the indefinite integral at least at an intermediate level. Thus, dummies should first read the lesson Not.
2) Be able to apply the Newton-Leibniz formula and calculate the definite integral. Set up warm friendly relations with definite integrals can be found on the page Definite integral. Examples of solutions.
In fact, in order to find the area of a figure, you don’t need that much knowledge of the indefinite and definite integral. The task “calculate the area using a definite integral” always involves constructing a drawing, so much more topical issue will be your knowledge and skills in drawing. In this regard, it is useful to refresh your memory of the graphs of basic elementary functions, and, at a minimum, to be able to construct a straight line, parabola and hyperbola. This can be done (for many, it is necessary) using methodological material and articles on geometric transformations of graphs.
Actually, everyone has been familiar with the task of finding the area using a definite integral since school, and we will not go much further than the school curriculum. This article might not have existed at all, but the fact is that the problem occurs in 99 cases out of 100, when a student suffers from a hated school and enthusiastically masters a course in higher mathematics.
The materials of this workshop are presented simply, in detail and with a minimum of theory.
Let's start with a curved trapezoid.
Curvilinear trapezoid is a flat figure bounded by an axis, straight lines, and the graph of a function continuous on an interval that does not change sign on this interval. Let this figure be located not lower x-axis:
Then the area of a curvilinear trapezoid is numerically equal to a definite integral. Any definite integral (that exists) has a very good geometric meaning. In class Definite integral. Examples of solutions I said that a definite integral is a number. And now it’s time to state one more useful fact. From the point of view of geometry, the definite integral is AREA.
That is, the definite integral (if it exists) geometrically corresponds to the area of a certain figure. For example, consider the definite integral. The integrand defines a curve on the plane located above the axis (those who wish can make a drawing), and the definite integral itself is numerically equal to the area of the corresponding curvilinear trapezoid.
Example 1
This is a typical assignment statement. First and the most important moment solutions - drawing. Moreover, the drawing must be constructed RIGHT.
When constructing a drawing, I recommend the following order: at first it is better to construct all straight lines (if they exist) and only Then– parabolas, hyperbolas, graphs of other functions. It is more profitable to build graphs of functions point by point, the point-by-point construction technique can be found in the reference material Graphs and properties of elementary functions. There you can also find very useful material for our lesson - how to quickly build a parabola.
In this problem, the solution might look like this.
Let's draw the drawing (note that the equation defines the axis):
I will not shade the curved trapezoid; it is obvious here what area we are talking about. The solution continues like this:
On the segment, the graph of the function is located above the axis, That's why:
Answer:
Who has difficulties with calculating the definite integral and applying the Newton-Leibniz formula 
, refer to the lecture Definite integral. Examples of solutions.
After the task is completed, it is always useful to look at the drawing and figure out whether the answer is real. In this case, we count the number of cells in the drawing “by eye” - well, there will be about 9, it seems to be true. It is completely clear that if we got, say, the answer: 20 square units, then it is obvious that a mistake was made somewhere - 20 cells clearly do not fit into the figure in question, at most a dozen. If the answer is negative, then the task was also solved incorrectly.
Example 2
Calculate the area of a figure bounded by lines , , and axis
This is an example for independent decision. Full solution and answer at the end of the lesson.
What to do if the curved trapezoid is located under the axle?
Example 3
Calculate the area of the figure bounded by lines and coordinate axes.
Solution: Let's make a drawing: 
If a curved trapezoid is located under the axle(or at least no higher given axis), then its area can be found using the formula:
In this case: 
Attention! The two types of tasks should not be confused:
1) If you are asked to solve simply a definite integral without any geometric meaning, then it may be negative.
2) If you are asked to find the area of a figure using a definite integral, then the area is always positive! That is why the minus appears in the formula just discussed.
In practice, most often the figure is located in both the upper and lower half-plane, and therefore, from the simplest school problems we move on to more meaningful examples.
Example 4
Find the area of a plane figure bounded by the lines , .
Solution: First you need to complete the drawing. Generally speaking, when constructing a drawing in area problems, we are most interested in the points of intersection of lines. Let's find the intersection points of the parabola and the straight line. This can be done in two ways. The first method is analytical. We solve the equation:
This means that the lower limit of integration is , the upper limit of integration is .
If possible, it is better not to use this method..
It is much more profitable and faster to construct lines point by point, and the limits of integration become clear “by themselves.” The point-by-point construction technique for various graphs is discussed in detail in the help Graphs and properties of elementary functions. Nevertheless, the analytical method of finding limits still sometimes has to be used if, for example, the graph is large enough, or the detailed construction did not reveal the limits of integration (they can be fractional or irrational). And we will also consider such an example.
Let's return to our task: it is more rational to first construct a straight line and only then a parabola. Let's make the drawing: 
I repeat that when constructing pointwise, the limits of integration are most often found out “automatically”.
And now the working formula: If there is some continuous function on the segment greater than or equal to some continuous function , then the area of the figure bounded by the graphs of these functions and the lines , , can be found using the formula: 
Here you no longer need to think about where the figure is located - above the axis or below the axis, and, roughly speaking, it matters which graph is HIGHER(relative to another graph), and which one is BELOW.
In the example under consideration, it is obvious that on the segment the parabola is located above the straight line, and therefore it is necessary to subtract from
The completed solution might look like this:
The desired figure is limited by a parabola above and a straight line below.
On the segment, according to the corresponding formula:
Answer:
In fact school formula for the area of a curvilinear trapezoid in the lower half-plane (see simple example No. 3) - a special case of the formula 
. Since the axis is specified by the equation, and the graph of the function is located no higher axes, then 
And now a couple of examples for your own solution
Example 5
Example 6
Find the area of the figure bounded by the lines , .
When solving problems involving calculating area using a definite integral, a funny incident sometimes happens. The drawing was done correctly, the calculations were correct, but due to carelessness... the area of the wrong figure was found, this is exactly how your humble servant screwed up several times. Here real case from life:
Example 7
Calculate the area of the figure bounded by the lines , , , .
Solution: First, let's make a drawing: 
...Eh, the drawing came out crap, but everything seems to be legible.
The figure whose area we need to find is shaded blue(look carefully at the condition - how the figure is limited!). But in practice, due to inattention, a “glitch” often arises that you need to find the area of a figure that is shaded green!
This example is also useful in that it calculates the area of a figure using two definite integrals. Really:
1) On the segment above the axis there is a graph of a straight line;
2) On the segment above the axis there is a graph of a hyperbola.
It is quite obvious that the areas can (and should) be added, therefore:
Answer:
Let's move on to another meaningful task.
Example 8
Calculate the area of a figure bounded by lines,
Let’s present the equations in “school” form and make a point-by-point drawing: 
From the drawing it is clear that our upper limit is “good”: .
But what is the lower limit?! It is clear that this is not an integer, but what is it? May be ? But where is the guarantee that the drawing is made with perfect accuracy, it may well turn out that... Or the root. What if we built the graph incorrectly?
In such cases you have to spend extra time and clarify the limits of integration analytically.
Let's find the intersection points of a straight line and a parabola.
To do this, we solve the equation: 
,
Really, .
The further solution is trivial, the main thing is not to get confused in substitutions and signs; the calculations here are not the simplest.
On the segment 
, according to the corresponding formula: 
Answer: 
Well, to conclude the lesson, let’s look at two more difficult tasks.
Example 9
Calculate the area of the figure bounded by the lines , ,
Solution: Let's depict this figure in the drawing. 
Damn, I forgot to sign the schedule, and, sorry, I didn’t want to redo the picture. Not a drawing day, in short, today is the day =)
For point-by-point construction you need to know appearance sinusoids (and generally useful to know graphs of all elementary functions), as well as some sine values, they can be found in trigonometric table. In some cases (as in this case), it is possible to construct a schematic drawing, on which the graphs and limits of integration should be fundamentally correctly displayed.
There are no problems with the limits of integration here; they follow directly from the condition: “x” changes from zero to “pi”. Let's make a further decision:
On the segment, the graph of the function is located above the axis, therefore:
Example1 . Calculate the area of the figure bounded by the lines: x + 2y – 4 = 0, y = 0, x = -3, and x = 2
Let's construct a figure (see figure) We construct a straight line x + 2y – 4 = 0 using two points A(4;0) and B(0;2). Expressing y through x, we get y = -0.5x + 2. Using formula (1), where f(x) = -0.5x + 2, a = -3, b = 2, we find
S = = [-0.25=11.25 sq. units
Example 2. Calculate the area of the figure bounded by the lines: x – 2y + 4 = 0, x + y – 5 = 0 and y = 0.
Solution. Let's construct the figure.
Let's construct a straight line x – 2y + 4 = 0: y = 0, x = - 4, A(-4; 0); x = 0, y = 2, B(0; 2).
Let's construct a straight line x + y – 5 = 0: y = 0, x = 5, C(5; 0), x = 0, y = 5, D(0; 5).
Let's find the point of intersection of the lines by solving the system of equations:
x = 2, y = 3; M(2; 3).
To calculate the required area, we divide the triangle AMC into two triangles AMN and NMC, since when x changes from A to N, the area is limited by a straight line, and when x changes from N to C - by a straight line
For triangle AMN we have: ; y = 0.5x + 2, i.e. f(x) = 0.5x + 2, a = - 4, b = 2.
For triangle NMC we have: y = - x + 5, i.e. f(x) = - x + 5, a = 2, b = 5.
By calculating the area of each triangle and adding the results, we find:
sq. units
sq. units
9 + 4, 5 = 13.5 sq. units Check: = 0.5AC = 0.5 sq. units
Example 3. Calculate the area of a figure bounded by lines: y = x 2 , y = 0, x = 2, x = 3.
In this case, you need to calculate the area of a curved trapezoid bounded by the parabola y = x 2 , straight lines x = 2 and x = 3 and the Ox axis (see figure) Using formula (1) we find the area of the curvilinear trapezoid
= = 6 sq. units
Example 4. Calculate the area of the figure bounded by the lines: y = - x 2 + 4 and y = 0
Let's construct the figure. The required area is enclosed between the parabola y = - x 2 + 4 and the Ox axis.
Let's find the intersection points of the parabola with the Ox axis. Assuming y = 0, we find x = Since this figure is symmetrical about the Oy axis, we calculate the area of the figure located to the right of the Oy axis, and double the result obtained: = +4x]sq. units 2 = 2 sq. units
Example 5. Calculate the area of a figure bounded by lines: y 2 = x, yx = 1, x = 4
Here you need to calculate the area of a curvilinear trapezoid bounded by the upper branch of the parabola 2 = x, Ox axis and straight lines x = 1 and x = 4 (see figure)
According to formula (1), where f(x) = a = 1 and b = 4, we have = (= sq. units.
Example 6 . Calculate the area of the figure bounded by the lines: y = sinx, y = 0, x = 0, x= .
The required area is limited by the half-wave of the sinusoid and the Ox axis (see figure).
We have - cosx = - cos = 1 + 1 = 2 sq. units
Example 7. Calculate the area of the figure bounded by the lines: y = - 6x, y = 0 and x = 4.
The figure is located under the Ox axis (see figure).
Therefore, we find its area using formula (3)
= =
Example 8. Calculate the area of the figure bounded by the lines: y = and x = 2. Construct the y = curve from the points (see figure). Thus, we find the area of the figure using formula (4)
Example 9 .
X 2 + y 2 = r 2 .
Here you need to calculate the area enclosed by the circle x 2 + y 2 = r 2 , i.e. the area of a circle of radius r with the center at the origin. Let's find the fourth part of this area by taking the limits of integration from 0
before; we have: 1 = = [
Hence, 1 =
Example 10. Calculate the area of a figure bounded by lines: y= x 2 and y = 2x
This figure is limited by the parabola y = x 2 and the straight line y = 2x (see figure) To determine the intersection points of the given lines, we solve the system of equations: x 2 – 2x = 0 x = 0 and x = 2
Using formula (5) to find the area, we obtain
= graph of a function y=x 2 +2 located above the axis Ox , That's why:
Answer: S =9 sq. units
After the task is completed, it is always useful to look at the drawing and figure out whether the answer is real. In this case, “by eye” we count the number of cells in the drawing - well, there will be about 9, it seems to be true. It is absolutely clear that if we got, say, the answer: 20 square units, then it is obvious that a mistake was made somewhere - 20 cells obviously do not fit into the figure in question, at most a dozen. If the answer is negative, then the task was also solved incorrectly.
What to do if the curved trapezoid is located under the axle Oh?
b) Calculate the area of a figure bounded by lines y=-e x , x=1 and coordinate axes.
Solution.
Let's make a drawing.
If a curved trapezoid completely located under the axis Oh , then its area can be found using the formula:
Answer: S=(e-1) sq. units" 1.72 sq. units
Attention! The two types of tasks should not be confused:
1) If you are asked to solve simply a definite integral without any geometric meaning, then it may be negative.
2) If you are asked to find the area of a figure using a definite integral, then the area is always positive! That is why the minus appears in the formula just discussed.
In practice, most often the figure is located in both the upper and lower half-plane.
With) Find the area of a plane figure bounded by lines y=2x-x 2, y=-x.
Solution.
First you need to complete the drawing. Generally speaking, when constructing a drawing in area problems, we are most interested in the points of intersection of lines. Let's find the intersection points of the parabola 
and straight 
This can be done in two ways. The first method is analytical.
We solve the equation:
This means that the lower limit of integration a=0 , upper limit of integration b=3 .
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We build the given lines: 1. Parabola - vertex at point (1;1); axis intersection Oh - points (0;0) and (0;2). 2. Straight line - bisector of the 2nd and 4th coordinate angles. And now Attention! If on the segment [ a;b] some continuous function f(x) greater than or equal to some continuous function g(x), then the area of the corresponding figure can be found using the formula: And it doesn’t matter where the figure is located - above the axis or below the axis, but what matters is which graph is HIGHER (relative to another graph) and which is BELOW. In the example under consideration, it is obvious that on the segment the parabola is located above the straight line, and therefore it is necessary to subtract from |
.You can construct lines point by point, and the limits of integration become clear “by themselves.” Nevertheless, the analytical method of finding limits still sometimes has to be used if, for example, the graph is large enough, or the detailed construction did not reveal the limits of integration (they can be fractional or irrational).
The desired figure is limited by a parabola above and a straight line below.
On the segment 
, according to the corresponding formula:
Answer: S =4.5 sq. units
Problem 1(about calculating the area of a curved trapezoid).
In the Cartesian rectangular coordinate system xOy, a figure is given (see figure) bounded by the x axis, straight lines x = a, x = b (a curved trapezoid. It is required to calculate the area of the curved trapezoid.
Solution. Geometry gives us recipes for calculating the areas of polygons and some parts of a circle (sector, segment). Using geometric considerations, we can only find an approximate value of the required area, reasoning as follows.
Let's split the segment [a; b] (base of a curved trapezoid) into n equal parts; this partition is carried out using points x 1, x 2, ... x k, ... x n-1. Let us draw straight lines through these points parallel to the y-axis. Then the given curvilinear trapezoid will be divided into n parts, into n narrow columns. The area of the entire trapezoid is equal to the sum of the areas of the columns.
Let us consider the k-th column separately, i.e. a curved trapezoid whose base is a segment. Let's replace it with a rectangle with the same base and height equal to f(x k) (see figure). The area of the rectangle is equal to \(f(x_k) \cdot \Delta x_k \), where \(\Delta x_k \) is the length of the segment; It is natural to consider the resulting product as an approximate value of the area of the kth column. 
If we now do the same with all the other columns, we will come to the following result: the area S of a given curvilinear trapezoid is approximately equal to the area S n of a stepped figure made up of n rectangles (see figure):
\(S_n = f(x_0)\Delta x_0 + \dots + f(x_k)\Delta x_k + \dots + f(x_(n-1))\Delta x_(n-1) \)
Here, for the sake of uniformity of notation, we assume that a = x 0, b = x n; \(\Delta x_0 \) - length of the segment, \(\Delta x_1 \) - length of the segment, etc.; in this case, as we agreed above, \(\Delta x_0 = \dots = \Delta x_(n-1) \)
So, \(S \approx S_n \), and this approximate equality is more accurate, the larger n.
By definition, it is believed that the required area of a curvilinear trapezoid is equal to the limit of the sequence (S n):
$$ S = \lim_(n \to \infty) S_n $$
Problem 2(about moving a point)
A material point moves in a straight line. The dependence of speed on time is expressed by the formula v = v(t). Find the movement of a point over a period of time [a; b].
Solution. If the movement were uniform, then the problem would be solved very simply: s = vt, i.e. s = v(b-a). For uneven movement, you have to use the same ideas on which the solution to the previous problem was based.
1) Divide the time interval [a; b] into n equal parts.
2) Consider a period of time and assume that during this period of time the speed was constant, the same as at time t k. So we assume that v = v(t k).
3) Let’s find the approximate value of the point’s movement over a period of time; we’ll denote this approximate value as s k
\(s_k = v(t_k) \Delta t_k \)
4) Find the approximate value of displacement s:
\(s \approx S_n \) where
\(S_n = s_0 + \dots + s_(n-1) = v(t_0)\Delta t_0 + \dots + v(t_(n-1)) \Delta t_(n-1) \)
5) The required displacement is equal to the limit of the sequence (S n):
$$ s = \lim_(n \to \infty) S_n $$
Let's summarize. Solutions various tasks reduced to the same mathematical model. Many problems from various fields of science and technology lead to the same model in the process of solution. So this mathematical model need to be specially studied.
The concept of a definite integral
Let us give a mathematical description of the model that was built in the three considered problems for the function y = f(x), continuous (but not necessarily non-negative, as was assumed in the considered problems) on the interval [a; b]:
1) split the segment [a; b] into n equal parts;
2) make up the sum $$ S_n = f(x_0)\Delta x_0 + f(x_1)\Delta x_1 + \dots + f(x_(n-1))\Delta x_(n-1) $$
3) calculate $$ \lim_(n \to \infty) S_n $$
In the know mathematical analysis it has been proven that this limit exists in the case of a continuous (or piecewise continuous) function. They call him a certain integral of the function y = f(x) over the segment [a; b] and denoted as follows:
\(\int\limits_a^b f(x) dx \)
The numbers a and b are called the limits of integration (lower and upper, respectively).
Let's return to the tasks discussed above. The definition of area given in Problem 1 can now be rewritten as follows:
\(S = \int\limits_a^b f(x) dx \)
here S is the area of the curvilinear trapezoid shown in the figure above. This is geometric meaning of a definite integral.
The definition of the displacement s of a point moving in a straight line with a speed v = v(t) over the period of time from t = a to t = b, given in Problem 2, can be rewritten as follows:
Newton - Leibniz formula
First, let's answer the question: what is the connection between the definite integral and the antiderivative?
The answer can be found in Problem 2. On the one hand, the displacement s of a point moving in a straight line with a speed v = v(t) over the period of time from t = a to t = b is calculated by the formula
\(S = \int\limits_a^b v(t) dt \)
On the other hand, the coordinate of a moving point is an antiderivative for speed - let's denote it s(t); this means that the displacement s is expressed by the formula s = s(b) - s(a). As a result we get:
\(S = \int\limits_a^b v(t) dt = s(b)-s(a) \)
where s(t) is the antiderivative of v(t).
The following theorem was proven in the course of mathematical analysis.
Theorem. If the function y = f(x) is continuous on the interval [a; b], then the formula is valid
\(S = \int\limits_a^b f(x) dx = F(b)-F(a) \)
where F(x) is the antiderivative of f(x).
The given formula is usually called Newton-Leibniz formula in honor of the English physicist Isaac Newton (1643-1727) and the German philosopher Gottfried Leibniz (1646-1716), who received it independently of each other and almost simultaneously.
In practice, instead of writing F(b) - F(a), they use the notation \(\left. F(x)\right|_a^b \) (it is sometimes called double substitution) and, accordingly, rewrite the Newton-Leibniz formula in this form:
\(S = \int\limits_a^b f(x) dx = \left. F(x)\right|_a^b \)
When calculating a definite integral, first find the antiderivative, and then carry out a double substitution.
Based on the Newton-Leibniz formula, we can obtain two properties of the definite integral.
Property 1. The integral of the sum of functions is equal to the sum of the integrals:
\(\int\limits_a^b (f(x) + g(x))dx = \int\limits_a^b f(x)dx + \int\limits_a^b g(x)dx \)
Property 2. The constant factor can be taken out of the integral sign:
\(\int\limits_a^b kf(x)dx = k \int\limits_a^b f(x)dx \)
Calculating the areas of plane figures using a definite integral
Using the integral, you can calculate the areas not only of curvilinear trapezoids, but also of flat figures more complex type, for example the one shown in the figure. The figure P is limited by straight lines x = a, x = b and graphs of continuous functions y = f(x), y = g(x), and on the segment [a; b] the inequality \(g(x) \leq f(x) \) holds. To calculate the area S of such a figure, we will proceed as follows:
\(S = S_(ABCD) = S_(aDCb) - S_(aABb) = \int\limits_a^b f(x) dx - \int\limits_a^b g(x) dx = \)
\(= \int\limits_a^b (f(x)-g(x))dx \)
So, the area S of a figure bounded by straight lines x = a, x = b and graphs of functions y = f(x), y = g(x), continuous on the segment and such that for any x from the segment [a; b] the inequality \(g(x) \leq f(x) \) is satisfied, calculated by the formula
\(S = \int\limits_a^b (f(x)-g(x))dx \)
Table of indefinite integrals (antiderivatives) of some functions
$$ \int 0 \cdot dx = C $$ $$ \int 1 \cdot dx = x+C $$ $$ \int x^n dx = \frac(x^(n+1))(n+1 ) +C \;\; (n \neq -1) $$ $$ \int \frac(1)(x) dx = \ln |x| +C $$ $$ \int e^x dx = e^x +C $$ $$ \int a^x dx = \frac(a^x)(\ln a) +C \;\; (a>0, \;\; a \neq 1) $$ $$ \int \cos x dx = \sin x +C $$ $$ \int \sin x dx = -\cos x +C $$ $ $ \int \frac(dx)(\cos^2 x) = \text(tg) x +C $$ $$ \int \frac(dx)(\sin^2 x) = -\text(ctg) x +C $$ $$ \int \frac(dx)(\sqrt(1-x^2)) = \text(arcsin) x +C $$ $$ \int \frac(dx)(1+x^2 ) = \text(arctg) x +C $$ $$ \int \text(ch) x dx = \text(sh) x +C $$ $$ \int \text(sh) x dx = \text(ch ) x +C $$Let the function be non-negative and continuous on the interval. Then, according to geometric sense of a certain integral, the area of a curvilinear trapezoid bounded above by the graph of this function, below by the axis, on the left and right by straight lines and (see Fig. 2) is calculated by the formula
Example 9. Find the area of a figure bounded by a line 
and axis.
Solution. Function graph 
is a parabola whose branches are directed downward. Let's build it (Fig. 3). To determine the limits of integration, we find the points of intersection of the line (parabola) with the axis (straight line). To do this, we solve the system of equations
We get: 
, where , ; hence, , .
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We find the area of the figure using formula (5):
If the function is non-positive and continuous on the segment , then the area of the curvilinear trapezoid bounded below by the graph of this function, above by the axis, on the left and right by straight lines and , is calculated by the formula
. (6)
If the function is continuous on a segment and changes sign at a finite number of points, then the area of the shaded figure (Fig. 4) is equal to the algebraic sum of the corresponding definite integrals:
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Example 10. Calculate the area of the figure bounded by the axis and the graph of the function at .
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Solution. Let's make a drawing (Fig. 5). The required area is the sum of the areas and . Let's find each of these areas. First, we determine the limits of integration by solving the system 
We get , . Hence:
;
.
Thus, the area of the shaded figure is
(sq. units).
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Finally, let the curvilinear trapezoid be bounded above and below by the graphs of functions continuous on the segment and ,
and on the left and right - straight lines and (Fig. 6). Then its area is calculated by the formula
. (8)
Example 11. Find the area of the figure bounded by the lines and.
Solution. This figure is shown in Fig. 7. Let's calculate its area using formula (8). Solving the system of equations we find, ; hence, , . On the segment we have: . This means that in formula (8) we take as x, and as a quality – . We get:
(sq. units).
More complex problems of calculating areas are solved by dividing the figure into non-overlapping parts and calculating the area of the entire figure as the sum of the areas of these parts.
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Example 12. Find the area of the figure bounded by the lines , , .
Solution. Let's make a drawing (Fig. 8). This figure can be considered as a curvilinear trapezoid, bounded from below by the axis, to the left and right - by straight lines and, from above - by graphs of functions and. Since the figure is limited from above by the graphs of two functions, to calculate its area, we divide this straight line figure into two parts (1 is the abscissa of the point of intersection of the lines and ). The area of each of these parts is found using formula (4):
(sq. units); 
(sq. units). Hence:
(sq. units).
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In conclusion, we note that if a curvilinear trapezoid is limited by straight lines and , axis and continuous on the curve (Fig. 9), then its area is found by the formula
Volume of a body of rotation
Let a curvilinear trapezoid, bounded by the graph of a function continuous on a segment, axis , straight lines and , rotate around the axis (Fig. 10). Then the volume of the resulting body of rotation is calculated by the formula
. (9)
Example 13. Calculate the volume of a body obtained by rotating around the axis of a curvilinear trapezoid bounded by a hyperbola, straight lines, and axis.
Solution. Let's make a drawing (Fig. 11).
From the conditions of the problem it follows that , . From formula (9) we get
.
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Volume of a body obtained by rotation around an axis Oh curvilinear trapezoid bounded by straight lines y = c And y = d, axis Oh and a graph of a function continuous on a segment (Fig. 12), determined by the formula
. (10)
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Example 14. Calculate the volume of a body obtained by rotating around an axis Oh curvilinear trapezoid bounded by lines X 2 = 4at, y = 4, x = 0 (Fig. 13).
Solution. In accordance with the conditions of the problem, we find the limits of integration: , . Using formula (10) we obtain:
Rice. 13
Arc length of a plane curve
Let the curve given by the equation , where , lie in the plane (Fig. 14).
Rice. 14
Definition. The length of an arc is understood as the limit to which the length of a broken line inscribed in this arc tends, when the number of links of the broken line tends to infinity, and the length of the largest link tends to zero.
If a function and its derivative are continuous on the segment, then the arc length of the curve is calculated by the formula
. (11)
Example 15. Calculate the arc length of the curve enclosed between the points for which 
.
Solution. From the problem conditions we have 
. Using formula (11) we obtain:
.
4. Improper integrals
with infinite limits of integration
When introducing the concept of a definite integral, it was assumed that the following two conditions were satisfied:
a) limits of integration A and are finite;
b) the integrand is bounded on the interval.
If at least one of these conditions is not satisfied, then the integral is called not your own.
Let us first consider improper integrals with infinite limits of integration.
Definition. Let the function be defined and continuous on the interval, then and unlimited on the right (Fig. 15).
If the improper integral converges, then this area is finite; if the improper integral diverges, then this area is infinite.
Rice. 15
An improper integral with an infinite lower limit of integration is defined similarly:
. (13)
This integral converges if the limit on the right side of equality (13) exists and is finite; otherwise the integral is said to be divergent.
An improper integral with two infinite limits of integration is defined as follows:
, (14)
where с is any point of the interval. The integral converges only if both integrals on the right side of equality (14) converge.
;G) 
= [select a complete square in the denominator: ] = 
[replacement:
] = 
This means that the improper integral converges and its value is equal to .
