Calculating the areas of figures using integrals. Lesson topic: "Calculating the areas of plane figures using a definite integral" Brief theoretical information

Oral work 1. Using the integral, express the areas of the figures shown in the figures:

2. Calculate the integrals:

Find the area of ​​the figure:

5)1/3; ln2 ;√2

A little history

"Integral" invented Jacob Bernoulli(1690)

"to restore" from Latin integro

"whole" from Latin integer

"Primitive function"

from Latin

primitivus– initial,

Joseph Louis Lagrange

Integral in antiquity

The first known method for calculating integrals is Eudoxus exhaustion method (approximately 370 BC BC), who tried to find areas and volumes by breaking them into an infinite number of parts for which the area or volume was already known.

This method was picked up and developed Archimedes , and was used to calculate the areas of parabolas and approximate the area of ​​a circle.

Eudoxus of Cnidus

Isaac Newton (1643-1727)

The most complete presentation of differential and integral calculus is contained in

Variables - fluents (antiderivative or indefinite integral)

Rate of change of fluent - fluxion (derivative)

Leibniz Gottfried Wilhelm (1646-1716)

• first used by Leibniz at the end

The symbol was formed from the letter

S - word abbreviations

summa(sum)

Formulas for calculating the areas of shaded figures in drawings

Algorithm for calculating the area of ​​a plane figure :

• According to the conditions of the task, make a schematic drawing.
• Represent the required function as the sum or difference of the areas of curvilinear trapezoid, select the appropriate formula.
• Find the limits of integration (a and b) from the conditions of the problem or drawing, if they are not specified.
• Calculate the area of ​​each curved trapezoid and the area of ​​the desired figure.

TASK

It was decided to plant a flowerbed in front of the school building. But the shape of the flowerbed should not be round, square or rectangular. It should contain straight and curved lines. Let it be a flat figure bounded by lines

Y = 4/X + 2; X = 4; Y = 6.

Let's calculate the area of ​​the resulting figure using the formula:

Where f(x)= 6 , A g(x)=4/x +2

Since 50 rubles are paid for each square meter, the earnings will be:

6.4 * 50 = 320 (rubles).

Homework:

Practical work on the topic: “Calculating the areas of plane figures using a definite integral”

Goal of the work: master the ability to solve problems involving calculating the area of ​​a curvilinear plane figure using a definite integral.

Equipment: instruction card, table of integrals, lecture material on the topic: “Definite integral. Geometric meaning of a definite integral."

Guidelines:

1) Study the lecture materials: “Definite integral. Geometric meaning of a definite integral."

Brief theoretical information

Definite integral of a function on the segment - this is the limit, to

to which the integral sum tends as the length of the largest partial segment tends to zero.

The lower limit of integration is the upper limit of integration.

To calculate a definite integral, use Newton's formula-

Leibniz:

Geometric meaning of the definite integral. If integrable on

segment, the function is non-negative, then it is numerically equal to the area of ​​the curvilinear trapezoid:

Curvilinear trapezoid - figure bounded by the graph of a function

Abscissa axis and straight lines, .

Various cases of arrangement of flat figures in the coordinate plane are possible:

If a curved trapezoid with a base is bounded below the curve , then from symmetry considerations it is clear that the area of ​​the figure is equal to or.

If a figure is bounded by a curve that takes both positive and negative values . In this case, in order to calculate the area of ​​the desired figure, it is necessary to divide it into parts, then

If a plane figure is bounded by two curves and , then its area can be found using the areas of two curvilinear trapezoids: and. In this case, the area of ​​the desired figure can be calculated using the formula:

Example. Calculate the area of ​​the figure bounded by the lines:

Solution. 1) Construct a parabola and a straight line in the coordinate plane (drawing for the problem).

2) Select (shade) the figure bounded by these lines.

Drawing for the problem

3) Find the abscissa of the intersection points of the parabola and the straight line. For this we will decide

system by comparison:

We find the area of ​​the figure as the difference between the areas of curvilinear trapezoids,

bounded by a parabola and a straight line.

5) Answer.

Algorithm for solving the problem of calculating the area of ​​a figure bounded by given lines:

Construct the given lines in one coordinate plane.

Shade the figure bounded by these lines.

Determine the limits of integration (find the abscissa of the intersection points of the curves).

Calculate the area of ​​the figure by choosing the required formula.

Write down the answer.

2) Do the following task according to one of the options:

Exercise. Calculate the area of ​​figures bounded by lines (use the algorithm for solving the problem of calculating the area of ​​a figure):

There are three lessons on this topic, this lesson is the second.

Lesson objectives:

Consolidating and deepening knowledge about the definite integral and its application to finding the area of ​​figures;

Formation of skills in applying knowledge and methods of action in changed and new learning situations; - development of information and communication culture of students;

Fostering cognitive activity, the ability to work in a team, perseverance and goal achievement.

Lesson objectives:

Repeat the table and rules for finding antiderivatives, the concept of a curvilinear trapezoid, the algorithm for finding the area of ​​a curvilinear trapezoid; - apply existing knowledge and skills to find the areas of plane figures.

Forms of organizing students' work: work in groups.

Equipment and programs used: interactive whiteboard Smart Board, “Live Mathematics”.

Interactive whiteboard software features used:

Function – curtain:

Function – cloning an object:

Function – dragging an object;

Function: smart pen.

Download:

Preview:

Lesson on the topic: “Calculating the areas of figures using integrals”

In 11th grade.

During the classes:

1. Organizing time ((readiness for the lesson is checked, the topic and purpose of the lesson are announced, the date is written down).

The lesson goes under the motto: Tell me and I will forget, Show me and I will remember, Let me act on my own, and I will learn.

Confucius.

1. Stage of updating previously acquired knowledge(the purpose of this stage: to repeat the table and rules for finding antiderivatives, the concept of a curvilinear trapezoid, the algorithm for finding the area of ​​a curvilinear trapezoid).

Teacher: In previous lessons we became acquainted with the concept of antiderivatives, with a table and the rules for finding them.

Question 1 : What is called the antiderivative for the function y = f (x) on a certain interval? Question 2 : How to set all antiderivative functions y = f (x) if F (x) is one of them? Question 3: List the rules for finding antiderivatives. After the students answer, slide 2 opens, the curtain moves back, behind which questions for students are hidden. Exercise 1 : Find one of the antiderivatives for the specified functions. (students use the drag-and-drop function to match the function and the antiderivative). Task 2 : For the specified function, find one of the antiderivatives whose graph passes through the given point. (Students decide independently on the spot; one of the students checks the answer by moving the screen).

A) Functions: 2x 5 – 3x 2; 3 cos x – 4 sin x; 3e x + 5 x – 2; e 2x – cos3x; 1/x + 1/ sin 2 x – x.

Antiderivatives: ln |x| -ctg x – x 2 /2; 1/2e 2x – 1/3 sin 3x; x 6 /3 – x 3; 3 sin x + 4 cos x; 3e x + 5 x /ln5.

B) For the function f (x) = 2x + 3, find an antiderivative whose graph passes through the point M (1;2).

Question 4: What figure is called a curved trapezoid? Task 3: Write down the missing condition in the definition written on the slide. Task 4: Write down Newton's Leibniz formula.

Task 5: Calculate the integral. (Students calculate independently, followed by verification). A) x 2 – 2x) dx; b)

Task 6: Calculate the area of ​​the figure bounded by the lines y = 0, x = e, y = 1/x. (Students independently complete the task and then check it by opening the screens on the board).

1. The stage of forming and practicing skills when solving various tasks on the topic “Calculating the areas of shapes using integrals»

1. Students remember the properties of areas

and give an example of a figure whose area can be calculated using the formula S =Calculate the area of ​​the figure bounded by the lines y = 0, y = x 2 – 4. (One student uses the smart pen function to write a solution on the interactive board).

2. Students discussplan for calculating the area of ​​a figure bounded by the lines y = x 2 – 6x +11 and y = x +1. Each stage is accompanied by the opening of the curtain.

1. Group work. The class is divided into groups in advance. Three students work at the board, and the rest of the students work in three options (groups are divided by option) on the spot:Calculate the area of ​​the figure bounded by the lines:Option 1 - y = (x – 3) 2 , y = 0, x = 1, x = 4. Option 2 – y = x – 2, y = x 2 - 4x +2. Option 3 – y = x, y = 5 – x, x =1, x = 2. Check after opening the screens.
2. Group work. For each of the next 8 slides you need to calculate the area of ​​the figure. Students in groups have a data set of drawings. Students choose a formula to find the area. A slide opens, to the right of the drawing there are formulas on which the cloning function is applied. After discussion in groups, one student from the group comes out and moves the selected formula or writes their own if there is not one on the board. Discussion follows: - Why was this formula chosen? - Are there other ways to find the area of ​​a given figure? - Which formula is most convenient to use?

Homework.

Lesson summary. Students answer the questions: - What was done in the lesson? - What new did they learn in the lesson? - How did they work in this group?

1125 Calculation of the areas of plane figures using the integral Methodological instructions for performing independent work in mathematics for 1st year students of the Faculty of Secondary Professional Education Compiled by S.L. Rybina, N.V. Fedotova 0 Ministry of Education and Science of the Russian Federation Federal State Budgetary Educational Institution of Higher Education "Voronezh State University of Architecture and Civil Engineering" Calculation of the areas of plane figures using the integral Guidelines for performing independent work in mathematics for 1st year students of the faculty SPO Compiled by S.L. Rybina, N.V. Fedotova Voronezh 2015 1 UDC 51:373(07) BBK 22.1ya721 Compiled by: Rybina S.L., Fedotova N.V. Calculation of the areas of plane figures using an integral: guidelines for performing independent work in mathematics for 1st year students of secondary vocational education/Voronezh State Agrarian University; comp.: S.L. Rybina, N.V. Fedotova. – Voronezh, 2015. – p. Theoretical information on calculating the areas of plane figures using the integral is given, examples of problem solving are given, and tasks for independent work are given. Can be used to prepare individual projects. Intended for 1st year students of the Faculty of Open Secondary Education. Il. 18. Bibliography: 5 titles. UDC 51:373(07) BBK 22.1я721 Published by decision of the educational and methodological council of the Voronezh State Agrarian University Reviewer – Glazkova Maria Yurievna, Ph.D. physics and mathematics Sciences, Associate Professor, Lecturer at the Department of Higher Mathematics, Voronezh State Agrarian University 2 Introduction These guidelines are intended for 1st year students of the Faculty of Secondary Professional Education of all specialties. Paragraph 1 provides theoretical information on calculating the areas of plane figures using an integral, paragraph 2 provides examples of solving problems, and paragraph 3 offers problems for independent work. General provisions Independent work of students is work that they perform on the instructions of the teacher, without his direct participation (but under his guidance) at a time specially provided for this. Goals and objectives of independent work: systematization and consolidation of acquired knowledge and practical skills of students; deepening and expanding theoretical and practical knowledge; developing the ability to use special reference literature and the Internet; development of students’ cognitive abilities and activity, creative initiative, independence, responsibility and organization; formation of independent thinking, abilities for self-development, self-improvement and self-realization; development of research knowledge. providing a knowledge base for professional training of graduates in accordance with the Federal State Educational Standard for Secondary Professional Education; formation and development of general competencies defined in the Federal State Educational Standard for Secondary Professional Education; preparation for the formation and development of professional competencies corresponding to the main types of professional activity. systematization, consolidation, deepening and expansion of the acquired theoretical knowledge and practical skills of students; development of cognitive abilities and activity of students: creative initiative, independence, responsibility and organization; formation of independent thinking: the ability for self-development, self-improvement and self-realization; mastering practical skills in using information and communication technologies in professional activities; development of research skills. The criteria for assessing the results of a student’s extracurricular independent work are: the student’s level of mastery of educational material; 3 the student’s ability to use theoretical knowledge when solving problems; validity and clarity of the response; design of the material in accordance with the requirements of the Federal State Educational Standard. 4 1. Calculation of the areas of plane figures using the integral 1. Reference material. 1.1. A curved trapezoid is a figure bounded from above by the graph of a continuous and non-negative function y=f(x), from below by a segment of the Ox axis, and from the sides by line segments x=a, x=b (Fig. 1) Fig. 1 The area of ​​a curved trapezoid can be calculated using a definite integral: b S f x dx F x b a F b (1) F a a 1.2. Let the function y=f(x) be continuous on a segment and take positive values ​​on this segment (Fig. 2). Then you need to divide the segment into parts, then calculate using formula (1) the areas corresponding to these parts, add the resulting areas. S = S1 + S2 c S b f x dx f x dx a (2) c Fig. 2 1.3. In the case where the continuous function f(x)< 0 на отрезке [а,b], для вычисления площади криволинейной трапеции следует использовать формулу: 5 b S f (x) dx (3) a Рис. 3 1.4. Рассмотрим случай, когда фигура ограничена графиками произвольных функций у =f(x) и у = g(x), графики которых пересекаются в точках с абсциссами а и b (а < b). Пусть эти функции непрерывны на и f(x)> g(x) over the entire interval (a; b). In this case, the area of ​​the figure is calculated by the formula y b S= (f (x) g (x))dx y=f(x) (4) a 1 a -1 O -1 b 1 y=g(x) x Fig. 4 1.5. Problems of calculating the areas of flat figures can be solved according to the following plan: 1) according to the conditions of the problem, make a schematic drawing; 2) represent the desired figure as the sum or difference of the areas of curvilinear trapezoids. From the conditions of the problem and the drawing, the limits of integration are determined for each component of the curvilinear trapezoid; 3) write each function in the form f x ; 4) calculate the area of ​​each curvilinear trapezoid and the desired figure. 6 2. Examples of solving problems 1. Calculate the area of ​​a curved trapezoid bounded by the lines y = x + 3, y = 0, x = 1 and x = 3. Solution: Let's draw the lines given by the equations and shade the curved trapezoid, the area of ​​which we will find. SАВД= Answer: 10. 2. The figure bounded by the lines y = -2x + 8, x = -1, y = 0 is divided by the line y = x2 – 4x + 5 into two parts. Find the area of ​​each part. Solution: Consider the function y = x2 – 4x +5. y = x2 – 4x +5 = (x2 – 4x + 4) – 4 + 5 = (x – 2)2 + 1, i.e. The graph of this function is a parabola with vertex K(2; 1). SABC= . 7 SABCME = S1 = SABCME + SEMC, S1 = S2 = SABC – S1, S2 = Answer: and = . . 3. Assignments for independent work Oral test 1. What figure is called a curved trapezoid? 2. Which of the figures are curved trapezoids: 3. How to find the area of ​​a curved trapezoid? 4. Find the area of ​​the shaded figure: 8 5. Name the formula for calculating the area of ​​the depicted figures: Written test 1. Which figure shows a figure that is not a curved trapezoid? 2. Using the Newton-Leibniz formula, calculate: A. Antiderivative of the function; B. Area of ​​a curved trapezoid; V. Integral; D. Derivative. 3. Find the area of ​​the shaded figure: 9 A. 0; B. –2; IN 1; D. 2. 4. Find the area of ​​the figure limited by the Ox axis and the parabola y = 9 – x2 A. 18; B. 36; V. 72; D. Cannot be calculated. 5. Find the area of ​​the figure bounded by the graph of the function y = sin x, the straight lines x = 0, x = 2 and the abscissa axis. A. 0; B. 2; AT 4; D. Cannot be calculated. Option 1 Calculate the area of ​​the figure bounded by the lines: a) y x2, b) y x2 c) y cos x, d) y 1, x3 y 0, x y 0; x, y 0, 0, 4; x x 1, x 0, x 6; 2. 10 Option 2 Calculate the area of ​​the figure bounded by the lines: b) y 1 2 x, y 2 x2 2 x, c) y sin x, d) y 1, x2 a) y y 0, x y 0; 0, x 0, x 3; 3 2, ; x 1. Option 3 Calculate the area of ​​the figure bounded by the lines: a) y = 2 – x3, y = 1, x = -1, x = 1; b) y = 5 – x2, y = 2x2 + 1, x = 0, x = 1; c) y = 2sin x, x = 0, x = p, y = 0; d) y = 2x – 2, y = 0, x = 3, x = 4. Option 4 Calculate the area of ​​the figure bounded by the lines: a) y = x2+1, y = 0, x = - 1, x = 2; b) y = 4 – x2 and y = x + 2; c) y = x2 + 2, y = 0, x = - 1, x = 2; d) y = 4 – x2 and y = 2 – x. Option 5 Calculate the area of ​​the figure bounded by the lines: a) y 7 x, x=3, x=5, y=0; b) y c) y d) y 8, x= - 8, x= - 4, y=0; x 0.5 x 2 4 x 10, y x 2; x 2, y x 6, x=-6 and coordinate axes. 11 Option 6 Calculate the area of ​​the figure bounded by the lines a) y 4 x 2, y = 0; b) y cos x, x, x c) y x 2 8 x 18, y d) y x, y 2, y=0; 2x 18; 1, x=4. x Option 7 Calculate the area of ​​the figure bounded by the lines a) y x 2 6 x, x = -1, x = 3, y = 0; b) y=-3x, x=1, x=2, y=0; c) y x 2 10 x 16, y=x+2; d) y 3 x, y = -x +4 and coordinate axes. Option 8 Calculate the area of ​​the figure bounded by the lines a) y sin x, x 3, x, y = 0; b) y x 2 4, x=-1, x=2, y=0; c) y x 2 2 x 3, y 3x 1; d) y x 2, y x 4 2, y = 0, Option 1 1. Calculate the area of ​​the figure bounded by the lines: a) y = x2, x = 1, x = 3, y = 0; b) y = 2cos x, y = 0, x = - Ï Ï , x= ; 2 2 c) y = 2x2, y = 2x. 2. (optional) Find the area of ​​the figure bounded by the graph of the function y = x2 – 2x + 3, tangent to the graph at its point with abscissa 2 and straight line x = -1. 12 Option 2 1. Calculate the area of ​​the figure bounded by the lines: a) y = x3, x = 1, x = 3, y = 0; b) y = 2cos x, y = 0, x = 0, x = Ï; 2 c) y = 0.5x2, y = x. 2. (optional) Find the area of ​​the figure bounded by the graph of the function y = 3 + 2x - x2, tangent to the graph at its point with abscissa 3 and straight line x = 0. Option 3 1. Calculate the area of ​​the figure bounded by the lines: a) y = x, x = 1, x = 2, y = 0; b) y = 2cos x, y = 0, x = Ï 3Ï , x= ; 2 2 c) y = x2, y = -x2 + 2. 2. (optional) Find the area of ​​the figure bounded by the graph of the function y = 2x - x2, tangent to the graph at its point with abscissa 2 and ordinate axis. Option 4 1. Calculate the area of ​​the figure bounded by the lines: a) y = 0.5 x, x = 1, x = 2, y = 0; b) y = 2cos x, y = 0, x = Ï Ï , x= ; 4 2 c) y = 9 - x2, y = 2x + 6. 2. (optional) Find the area of ​​the figure bounded by the graph of the function y = x2+ 2x, tangent to the graph at its point with abscissa -2 and ordinate axis. Tasks for working in pairs: 1. Calculate the area of ​​the shaded figure 2. Calculate the area of ​​the shaded figure 13 3. Calculate the area of ​​the shaded figure 4. Calculate the area of ​​the shaded figure 14 5. Calculate the area of ​​the shaded figure 6. Present the area of ​​the shaded figure as the sum or difference of the areas of curvilinear trapezoids bounded by the graphs of lines you know. 7. Imagine the area of ​​the shaded figure as the sum or difference of the areas of curvilinear trapezoids bounded by the graphs of the lines you know. 15 Bibliography 1. Sharygin, I. F. Mathematics: algebra and principles of mathematical analysis, geometry. Geometry. A basic level of. Grades 10 - 11: textbook / I.F. Sharygin. - 2nd ed., erased. – Moscow: Bustard, 2015. – 238 p. 2. Muravin G.K. Mathematics: algebra and principles of mathematical analysis, geometry. A basic level of. 11th grade: textbook / G.K. Muravin, O.V. Muravin - 2nd ed., erased. - Moscow: Bustard, 2015. - 189 p. 3. Muravin G.K. Mathematics: algebra and principles of mathematical analysis, geometry. A basic level of. 10th grade: textbook / G.K. Muravin, O.V. Muravina. - 2nd ed., erased. - Moscow: Bustard, 2013 – 285 p. 4. Studying geometry in grades 10-11: Method. recommendations for studies: Book. for teacher/S. M. Sahakyan, V. F. Butuzov. – 2nd ed. – M.: Education, 2014. – 222 p.: ill. 5. Study of algebra and beginnings of analysis in grades 10-11: Book. for the teacher / N. E. Fedorova, M. V. Tkacheva. – 2nd ed. – M.: Education, 2014. – 205 p.: ill. 6. Algebra and the beginnings of analysis. 10-11 grades: In two parts. Part 1: Textbook for general education. institutions / Mordkovich A.G. – 5th ed. – M.: Mnemosyne, 2014. – 375 p.: ill. Internet resources: 1. http://www.exponenta.ru/educat/links/l_educ.asp#0 – Useful links to mathematical and educational sites: Educational materials, tests 2. http://www.fxyz.ru / - Interactive reference book of formulas and information on algebra, trigonometry, geometry, physics. 3. http://maths.yfa1.ru - The reference book contains material on mathematics (arithmetic, algebra, geometry, trigonometry). 4. allmatematika.ru - Basic formulas in algebra and geometry: identity transformations, progressions, derivatives, stereometry, etc. 5. http://mathsun.ru/ – History of mathematics. Biographies of great mathematicians. 16 Contents Introduction. ........................................................ ........................................................ .................................. 3 Calculation of the areas of plane figures using the integral......... .................................... 5 1. Reference material........ ........................................................ ................................................. 5 2. Examples of problem solving................................................................. ........................................................ ......... 7 3. Tasks for independent work.................................... ............ .................................. 8 Bibliography.................. ........................................................ ........................................... 16 Calculating the areas of plane figures using the integral Guidelines for performing independent works in mathematics for 1st year students of the Faculty of Secondary Education Compiled by: Rybina Svetlana Leonidovna Fedotova Natalya Viktorovna Signed for printing __.__. 2015. Format 60x84 1/16. Academic ed. l. 1.1.Condition-bake l. 1.2. 394006, Voronezh, st. 20th anniversary of October, 84 17