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геометрический

смысл

определенного

интеграла

площадь

криволинейной

трапеции

высшая

математика

00:00:00

hello Andrey Gradient and

00:00:02

now higher mathematics

00:00:04

the topic of the lesson today is the geometric

00:00:07

meaning of a definite integral the area of a

00:00:10

curvilinear trapezoid

00:00:12

but before we begin to

00:00:16

consider the geometric meaning with you

00:00:18

I would like to formulate a very important

00:00:21

theorem, the so-called sufficient

00:00:25

condition for

00:00:26

the existence of a definite integral

00:00:28

or a sufficient condition for the integrability

00:00:30

of a function on a segment, so if a function y

00:00:37

is equal to f of x

00:00:38

continuously on a certain segment from a to

00:00:43

b, then a definite integral of this function

00:00:47

on this segment exists, this

00:00:51

statement is called a sufficient

00:00:52

condition for integration, pay

00:00:54

attention precisely enough and because

00:00:57

there are functions that are

00:01:01

discontinuous

00:01:02

but a definite integral from these functions

00:01:05

there are in particular these are functions that

00:01:10

have a finite number of discontinuity points on segments

00:01:14

and are limited on this segment

00:01:18

and so have remembered until a

00:01:22

sufficient condition for the existence of the

00:01:25

integral method is the continuity of the

00:01:28

integrand on the

00:01:31

integration segment so now let’s move on to the

00:01:34

geometric meaning, consider

00:01:37

the function,

00:01:43

consider the function y equals f from x y

00:01:53

is equal to f from x

00:01:56

let it let it be defined and

00:02:07

continuously continuous on some

00:02:15

segment

00:02:17

a.b. and on this segment it

00:02:26

is not negative and f from x is

00:02:31

greater than or equal to zero

00:02:33

for all x from the segment from a to b t cattle

00:02:40

d.b. let's

00:02:46

build a graph of this function in a Cartesian

00:02:49

rectangular coordinate system

00:02:54

[applause]

00:03:08

asics y-axis and

00:03:15

a graph of this function some curve

00:03:19

some beer and

00:03:26

consider the

00:03:28

segment both x axes

00:03:36

ab. let's draw vertical lines

00:03:42

x equals x equals b, well, more precisely, let's limit ourselves to

00:03:52

segments of these straight lines,

00:03:56

segments of

00:03:57

these straight lines, let's sign this y equals f

00:04:04

from x, here I'll sign the straight line x equals and this is

00:04:09

our straight line x equals b and pay attention

00:04:17

to the figure we see here a figure

00:04:23

bounded from above by the graph of a function y

00:04:28

equals f from x from

00:04:29

below it is limited by the x axis and from the side

00:04:35

it is limited by straight lines x equals a and x

00:04:39

equals b

00:04:40

such a figure is called a curvilinear

00:04:43

trapezoid, let's find the area of this

00:04:48

curvilinear trapezoid,

00:04:51

what we will do for this is divide

00:04:57

the segment a.b.

00:04:59

on n partial segments we will do this

00:05:04

using points x 0 x1 x2 and so on x n and in

00:05:10

such a way that . x zero

00:05:14

coincided with point a-a.

00:05:22

x n coincided with point b

00:05:26

and, accordingly, points x1 x2 and so

00:05:30

on x n minus 1 these are the points inside this

00:05:34

segment we develop in an arbitrary way in an

00:05:38

arbitrary way

00:05:40

x1

00:05:44

with 2 so on

00:05:54

x this is minus 1 x this is

00:06:06

x n minus 1 and so we broke it down let's

00:06:12

write down the steps and so on we find we

00:06:17

find the area of a curvilinear trapezoid

00:06:30

the area of a curvilinear trapezoid first

00:06:36

step 1st step

00:06:42

let's divide the segment Abe

00:06:50

into n partial segments segment Abe into m

00:06:59

partial segments well, how can I not

00:07:01

write down the hole all this was just

00:07:03

talked to you, so then in

00:07:09

each partial segment

00:07:11

we select a certain point c etc.

00:07:15

Well, that is, in the first segment c1 in the second

00:07:17

c2 and so on in the last cm we

00:07:20

randomly select points and

00:07:23

so on c1 c2

00:07:28

and so on on this segment all this and

00:07:31

on the last one. valuable and so we'll

00:07:45

take it. this is me from the segment and the cafe then

00:07:51

minus 1 x this is

00:07:53

where and where and we change from 1 to n 2 and

00:07:59

so on until n we took a point in an arbitrary

00:08:04

way and calculate it calculate f share of

00:08:13

this that is, for each value c we

00:08:17

calculate the value of the function value

00:08:19

functions, how to show this in pictures,

00:08:23

we find a point on the graph of socis and c and t

00:08:29

to each point c and you find a point on the

00:08:33

graph also 1

00:08:48

from 2 and so on with

00:08:51

this is valuable

00:09:06

on the graph,

00:09:11

well, it’s obvious that the ordinate is the ordinate of the

00:09:18

point on the graph with the abscissa c

00:09:22

this will be equal to f c this is yes, that is, I

00:09:26

pozzi this is the ordinate of the point of the

00:09:29

social graph at&t

00:09:32

in other words this is the length of the

00:09:35

corresponding segment between the x axis and the

00:09:39

graph of our function the graph of our

00:09:45

function so now the third step the third step let

00:09:50

's calculate the products let's calculate the

00:09:56

products of f from c and d multiply on

00:10:02

delta x this will explain what delta x is

00:10:06

where delta x is this is the difference between the

00:10:13

subsequent values of the previous one

00:10:15

give x this is minus x and up to -1 in other

00:10:21

words delta exito this is the length of the partial segment the

00:10:23

length of the partial

00:10:27

segment I remind us of their m partial

00:10:29

segments and so on what then is the

00:10:33

product of such a geometric

00:10:36

meaning of the product of f at&t by delta x

00:10:40

this is if we said delta x is delta

00:10:45

except this is the length of this partial

00:10:47

segment AB fathers this is the length of this

00:10:53

perpendicular and the product of

00:10:59

fc is by delta x this will be the

00:11:03

area of the rectangle with the base

00:11:07

delta x this and the height

00:11:10

about father this let's show this

00:11:14

rectangle with the height and father this

00:11:19

and the base delta x this let's

00:11:25

show this rectangle

00:11:46

look I showed a rectangle at the

00:11:53

base delta x this is the height of the f c this is the

00:11:57

product the area of such a

00:11:59

rectangle since such

00:12:02

products y us n yes why

00:12:06

because we have and changes from 1 to n

00:12:11

so the product will be n means

00:12:15

rectangles m and accordingly all

00:12:19

these products to the area of the

00:12:21

corresponding rectangles let's

00:12:22

show in the drawings

00:12:43

here ampere the rectangle corresponds to the

00:12:47

products of fac-1 by delta x 1 then

00:12:55

sometimes a rectangle the

00:13:10

second rectangle its area is equal to

00:13:13

f

00:13:14

let's and we have already written down fc2 by delta

00:13:20

x 2

00:13:22

and so on and so on yes we can show

00:13:48

so on the

00:14:04

next one

00:15:02

so we have depicted all such rectangles

00:15:07

now now the fourth step the fourth

00:15:11

step we will consider the sum of the father

00:15:23

one by delta x 1 plus fc2 to delta x

00:15:28

2 + and so on plus f from to delta x n

00:15:36

look here we

00:15:39

calculated all the products in the third step, now we

00:15:42

have summed everything up and with the product and

00:15:45

summed it up and

00:15:48

written it down compactly using the sum sign

00:15:51

fc this is me for delta x this and changes from 1

00:15:57

to n but let’s call this sum dreams dreams what

00:16:06

is this sum what

00:16:09

geometric sums does this sum mean

00:16:12

what henriksen each term of

00:16:15

this sum is the area of one such

00:16:19

rectangle if we add these

00:16:24

products

00:16:25

then accordingly their sum dreams

00:16:30

is the area of this stepped figure

00:16:34

that is a figure consisting of these

00:16:37

rectangles the

00:16:38

area of a stepped figure let's

00:16:41

write dreams this is the area the area of a

00:16:49

stepped

00:16:54

figure the area of a stepped figure

00:16:59

remember what our goal was to find the

00:17:02

area of a curvilinear trapezoid

00:17:04

look we can say yes that the area of a

00:17:07

curvilinear trapezoid with the designation c is

00:17:09

approximately approximately equal to the area of a

00:17:12

stepped figure let's write that

00:17:18

dreams are approximately is equal to c here y let's

00:17:23

denote the area of the curvilinear trapezoid

00:17:25

as c let's find the area and screen c reconcile

00:17:29

norves and this equality

00:17:35

anode approximate will be the more accurate it

00:17:41

will be the more accurate the shorter the length delta

00:17:56

x is the

00:17:59

smaller by delta x this is the more

00:18:02

definitely this equality will be more

00:18:04

accurate in

00:18:05

this way if we want to get the

00:18:07

area of attachment of the trapezoid, we must we

00:18:11

must direct delta x to zero, yes,

00:18:17

that is, the largest longest length

00:18:21

delta exito should tend to zero, but

00:18:23

accordingly the number of partitions should

00:18:26

tend to infinity and so we write the

00:18:33

fifth step the fifth step that the area of the

00:18:37

curvilinear trapezoid c equals

00:18:41

limit

00:18:42

c n and when n tends to infinity

00:18:47

n tends to infinity and the

00:18:51

greatest length delta x this tends

00:18:54

to zero

00:18:55

let's remember in the last lesson we

00:18:57

denoted the greatest length delta x this

00:19:00

by lambda, that is, lambda, in turn,

00:19:04

should tend to on

00:19:06

thus yes, it’s clear the area

00:19:08

trapezoid early to the limit of this partial

00:19:12

sum, that is, the limit of the area of the

00:19:14

stepped figure, so well, or

00:19:17

it will be equal to the limit n tends to

00:19:20

infinity to lambda tends to zero

00:19:26

fc this is

00:19:28

on delta x this and changes from 1 to n does not

00:19:32

change then you need to go to n and that we know

00:19:38

that this limit, this limit, by

00:19:43

definition, is a definite integral

00:19:46

of the function f of x on the segment a.b.

00:19:50

definite integral on the segment abe of the

00:19:53

function f of x g x thus

00:19:57

look at the area of a curvilinear

00:20:01

trapezoid is the limit of the area of a stepped

00:20:05

figure when the partition number n

00:20:08

tends to infinity and the main

00:20:11

base of our rectangles

00:20:15

tends to zero, that is, the clay

00:20:17

tends to zero in this case

00:20:19

the area tape 5 is a definite

00:20:21

integral, so the area of a trapezoid

00:20:26

is a definite integral of the function f

00:20:30

from x to the variable x on the segment a.b.

00:20:37

memorable yes where f from x

00:20:42

f from x is greater than or equal to zero on the segment

00:20:46

lunch for all x

00:20:48

from the segment a b a b area area of a

00:20:59

curvilinear trapezoid I remind you which is

00:21:07

limited from above by the graph of the function y

00:21:10

is equal to f from x reduced is limited by the x axis

00:21:14

on the left of the straight line x is equal to a vertical

00:21:17

straight line x is equal to and on the right of the vertical straight line

00:21:19

x is equal to b This is the

00:21:22

geometric meaning of a definite

00:21:25

integral of a certain interval,

00:21:27

this lesson is over if the video

00:21:31

was useful and interesting for you,

00:21:32

like it, subscribe to those who

00:21:34

have not subscribed, recommend the channel to friends and

00:21:36

acquaintances, I say goodbye to you until the next time to be continued

00:21:39

goodbye

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Геометрический смысл определенного интеграла. Площадь криволинейной трапеции. Высшая математика.

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