Antiderivative area of a curvilinear trapezoid. Online calculator. Calculate the definite integral (area of a curved trapezoid)

Let's consider a curved trapezoid bounded by the Ox axis, the curve y=f(x) and two straight lines: x=a and x=b (Fig. 85). Let's take an arbitrary value of x (just not a and not b). Let's give it an increment h = dx and consider a strip bounded by straight lines AB and CD, the Ox axis and the arc BD belonging to the curve under consideration. We will call this strip an elementary strip. The area of an elementary strip differs from the area of the rectangle ACQB by the curvilinear triangle BQD, and the area of the latter is less than the area of the rectangle BQDM with sides BQ = =h=dx) QD=Ay and area equal to hAy = Ay dx. As side h decreases, side Du also decreases and simultaneously with h tends to zero. Therefore, the area of the BQDM is second-order infinitesimal. The area of an elementary strip is the increment of the area, and the area of the rectangle ACQB, equal to AB-AC ==/(x) dx> is the differential of the area. Consequently, we find the area itself by integrating its differential. Within the figure under consideration, the independent variable l: changes from a to b, so the required area 5 will be equal to 5= \f(x) dx. (I) Example 1. Let us calculate the area bounded by the parabola y - 1 -x*, straight lines X =--Fj-, x = 1 and the O* axis (Fig. 86). at Fig. 87. Fig. 86. 1 Here f(x) = 1 - l?, the limits of integration are a = - and £ = 1, therefore J [*-t]\- -fl -- Г -1-±Л_ 1V1 -l-l-Ii-^ 3) |_ 2 3V 2 / J 3 24 24* Example 2. Let's calculate the area limited by the sinusoid y = sinXy, the Ox axis and the straight line (Fig. 87). Applying formula (I), we obtain A 2 S= J sinxdx= [-cos x]Q =0 -(-1) = lf Example 3. Calculate the area limited by the arc of the sinusoid ^у = sin jc, enclosed between two adjacent intersection points with the Ox axis (for example, between the origin and the point with the abscissa i). Note that from geometric considerations it is clear that this area will be twice the area of the previous example. However, let's do the calculations: I 5= | s\nxdx= [ - cosх)* - - cos i-(-cos 0)= 1 + 1 = 2. o Indeed, our assumption turned out to be correct. Example 4. Calculate the area bounded by the sinusoid and the Ox axis at one period (Fig. 88). Preliminary calculations suggest that the area will be four times larger than in Example 2. However, after performing the calculations, we obtain “i Г,*i S - \ sin x dx = [ - cos x]0 = = - cos 2l -(-cos 0) = - 1 + 1 = 0. This result requires clarification. To clarify the essence of the matter, we also calculate the area limited by the same sinusoid y = sin l: and the Ox axis in the range from l to 2i. Applying formula (I), we obtain 2l $2l sin xdx=[ - cosх]l = -cos 2i~)-c05i=- 1-1 =-2. Thus, we see that this area turned out to be negative. Comparing it with the area calculated in exercise 3, we find that their absolute values are the same, but the signs are different. If we apply property V (see Chapter XI, § 4), we get 2l I 2l J sin xdx= J sin * dx [ sin x dx = 2 + (- 2) = 0What happened in this example is not an accident. Always the area located below the Ox axis, provided that the independent variable changes from left to right, is obtained when calculated using integrals. In this course we will always consider areas without signs. Therefore, the answer in the example just discussed will be: the required area is 2 + |-2| = 4. Example 5. Let's calculate the area of the BAB shown in Fig. 89. This area is limited by the Ox axis, the parabola y = - xr and the straight line y - = -x+\. Area of a curvilinear trapezoid The required area OAB consists of two parts: OAM and MAV. Since point A is the intersection point of a parabola and a straight line, we will find its coordinates by solving the system of equations 3 2 Y = mx. (we only need to find the abscissa of point A). Solving the system, we find l; = ~. Therefore, the area has to be calculated in parts, first square. OAM and then pl. MAV: .... G 3 2, 3 G xP 3 1/2 U 2. QAM-^x 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

= area of the curved trapezoid formed by the function f, is equal to the increment of the antiderivative of this function:

Task 1:

Find the area of a curvilinear trapezoid bounded by the graph of the function: f(x) = x 2 and straight y = 0, x = 1, x = 2.

Solution: ( according to the algorithm slide 3)

Let's draw a graph of the function and lines

Let's find one of the antiderivatives of the function f(x) = x 2 :

Self-test on slide

Integral

Consider a curvilinear trapezoid defined by the function f on the segment [ a; b]. Let's break this segment into several parts. The area of the entire trapezoid will be divided into the sum of the areas of smaller curved trapezoids. ( slide 5). Each such trapezoid can be approximately considered a rectangle. The sum of the areas of these rectangles gives an approximate idea of the entire area of the curved trapezoid. The smaller we divide the segment [ a; b], the more accurately we calculate the area.

Let us write these arguments in the form of formulas.

Divide the segment [ a; b] into n parts by dots x 0 =a, x1,...,xn = b. Length k- th denote by xk = xk – xk-1. Let's make a sum

Geometrically, this sum represents the area of the figure shaded in the figure ( sh.m.)

Sums of the form are called integral sums for the function f. (sh.m.)

Integral sums give an approximate value of the area. The exact value is obtained by passing to the limit. Let's imagine that we are refining the partition of the segment [ a; b] so that the lengths of all small segments tend to zero. Then the area of the composed figure will approach the area of the curved trapezoid. We can say that the area of a curved trapezoid is equal to the limit of integral sums, Sc.t. (sh.m.) or integral, i.e.,

Definition:

Integral of a function f(x) from a to b called the limit of integral sums

= (sh.m.)

Newton-Leibniz formula.

We remember that the limit of integral sums is equal to the area of a curvilinear trapezoid, which means we can write:

Sc.t. = (sh.m.)

On the other hand, the area of a curved trapezoid is calculated using the formula

S k.t. (sh.m.)

Comparing these formulas, we get:

= (sh.m.)

This equality is called the Newton-Leibniz formula.

For ease of calculation, the formula is written as:

= = (sh.m.)

Tasks: (sh.m.)

1. Calculate the integral using the Newton-Leibniz formula: ( check on slide 5)

2. Compose integrals according to the drawing ( check on slide 6)

3. Find the area of the figure bounded by the lines: y = x 3, y = 0, x = 1, x = 2. ( Slide 7)

Finding the areas of plane figures ( slide 8)

How to find the area of figures that are not curved trapezoids?

Let two functions be given, the graphs of which you see on the slide . (sh.m.) Find the area of the shaded figure . (sh.m.). Is the figure in question a curved trapezoid? How can you find its area using the property of additivity of area? Consider two curved trapezoids and subtract the area of the other from the area of one of them ( sh.m.)

Let's create an algorithm for finding the area using animation on a slide:

Graph functions
Project the intersection points of the graphs onto the x-axis
Shade the figure obtained when the graphs intersect
Find curvilinear trapezoids whose intersection or union is the given figure.
Calculate the area of each of them
Find the difference or sum of areas

Oral task: How to obtain the area of a shaded figure (tell using animation, slide 8 and 9)

Homework: Work through the notes, No. 353 (a), No. 364 (a).

References

Algebra and the beginnings of analysis: a textbook for grades 9-11 of evening (shift) school / ed. G.D. Glaser. - M: Enlightenment, 1983.
Bashmakov M.I. Algebra and the beginnings of analysis: a textbook for 10-11 grades of secondary school / Bashmakov M.I. - M: Enlightenment, 1991.
Bashmakov M.I. Mathematics: textbook for institutions beginning. and Wednesday prof. education / M.I. Bashmakov. - M: Academy, 2010.
Kolmogorov A.N. Algebra and beginnings of analysis: textbook for grades 10-11. educational institutions / A.N. Kolmogorov. - M: Education, 2010.
Ostrovsky S.L. How to make a presentation for a lesson?/ S.L. Ostrovsky. – M.: First of September, 2010.

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