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Hello.

Our previous session, various We calculate the integral.

Regional differences in the definitions of examples that I have seen.

In addition, two kinds of problems we did.

B-type region was given.

And in which the boundaries of the integral we want to take the integral

According to him, we decide integrals used to calculate write.

Our fundamental principle in calculating the fine is there.

In multivariate functions, derivatives also, integration also available.

When two variables are variables temporarily one

We therefore freezes integral single we calculate an integral story.

However, the limits of this integral variable I might.

But, technically, an integral single storey are calculated.

In a second step these values, limits after placing

After an integral single storey happening again.

This time on the second floor is an integral limit values must be fixed.

I've seen a lot of my professional life iii wrong whether the variable values in the outermost

Although not used so that a certain in the frequency of encounters.

It should be noted that a significant

not the result of a lot of internal contradictions provides.

External borders should be fixed in the end he border

When using the values of a constant value Let's, let's find the number b.

Now here's the added torque on We will see examples.

Why on torque?

One important moments.

Most of these multi-ply integrally Where used

If you ask if where the analytical accounts This is the account of the moments.

This is the account of the area.

Areas with affordable, flat've done.

Areas on.

Here, too, the area of both the first

and the second moment calculations related to will do.

Their definition is not difficult.

There are great benefits in learning.

Because the physical sciences, statistics This also always moments in science

issues encountered and their calculation

an important gain in the know.

Furthermore, basic mathematics,

with infinitesimal in terms of fundamental analysis We're trying.

On the basis of the analysis of differentiation and integration the difference

distinguishes it from other branches of mathematics forever minors.

This is the limit of infinitely small are going to as you know.

As we have seen with these infinitesimal to consider an important skill.

This is also important in order to gain the skills moment of a media accounts.

For this reason, both useful in their own right useful

Calculation of the moment because most of of the issues encountered.

But they need to be calculated thinking skills to their win at

in terms of per mathematics, analysis a significant gain.

Therefore, at the moment we are working on.

Now easy to find the moments.

Now we have the first moment of area

can be easier to understand if we start bi concept.

What we say in the area?

Given an area to contact us.

Here we choose b x x plus delta x.

x, y plus delta y we choose.

Here is a small rectangular region occurs.

This delta x times the area of the rectangular region y is the delta.

J k because we are putting them There also must take equal.

Different values can be taken.

Indeed, in cases of complex technologies, designs

Not equal are they different.

Because of the critical region in bi-site must account more subtle.

In some places, functions very slow monotonic change.

Can be great values out there.

These are not our business at this stage.

But this, of getting them no different no difficulty in terms of accounts.

Summing this Delta approximately this region

limiting the area in which we find.

If we go to the limit delta x and delta

year zero in this field by taking certain we find the value.

Would be worth about earlier.

I got to know it.

Because it will do a lot of work on the computer, Do ks.

Especially interested in technology, social science, economics,

sosyolo, many who are interested in science will make such calculations.

Achieved when we go to the limit The results are integral to the call.

Here two floors if two sum We know that the integral.

D means that the integration of certain bi gives the area of the region.

These moments.

It is this point as the lever arms When you get these rectangles

in a small distance from the x axis thereof is there.

the y-axis is the distance.

If you think this is a mass this as leverage arms

y is the distance from the x axis y, y axis the distance x.

These small molecules, therefore, that the endless small to x-axis

Once this lever arm moment of the year

wherein the area of the mass times the mass of is going on, the product of the area.

Similarly, the y-axis by the torque

In this field, the lever arm x multiplication.

These are the same as this area ederkenki Summing up, and here we pass to the limit

d y times x times occurs and is an integral a integral occurs.

In the first moment of the science of statistics It is help you to find the average value.

The center of gravity in the physical sciences It is helpful to find.

The center of gravity of the area moments is obtained by dividing.

We field because we are a two-dimensional the area we work in the plane.

If you were going to be three-dimensional volume.

We will see him, but not here.

It will require the integration of three floors.

We're not ready for it yet again for him.

Here hunch something to be aware of it

It sounds like the opposite, but, considering the right that you find.

There are also factored in when m * y, m y When we got the x factor as well.

At first it sounds like the opposite of people but

because right at the moment according to the y-axis lever arm x.

the torque lever arm at the X axis by years.

For him that is coming.

But we want to find the center of gravity When we divide m a y

time here as you can see with X There entergral.

Do you like it here again, contrary to intuition

y from x, it was located in the center of gravity sounds like.

I need to pay attention to this.

The rationale for this, a little before the logic as I said.

Not surprising b-side.

The second moments of the exact same thought.

Again, this semi, small, racing, endless

this time to the region's small lever arm comes in the second degree.

According to the X-axis moment of inertia, or runner up

Order this infinitesimal moment of leverage with arm frame.

E, here the distance from y.

So the future of y squared.

According to the y-axis x lever arm this time.

In the second moment of it next frame.

Bi X, of course, also the product of y I might.

So he shaped body symmetry deviation shows.

In the examples we will see him.

This multiplying the x and y by multiplying the x and y'yl a magnitude obtained.

BI always works well now.

Coordinates relative to the center of teams The moment of inertia or second moment.

According to these centers by the lever arm away.

That means x squared plus y squared comes with it.

But already the integral of x i y square area

x, y square area is the integral of i x x these coordinates for

assembly according to the center of inertia The second moment, or torque.

Statistics variance is called in the language of their each.

It i x i y y X, the total becomes.

Yet perhaps the first foreboding here Reaction with our first

may seem to clash with our response i x x, i remark from inersiya

Coming y'yl going to square y.

i y y x square goes.

Apply the same logic as in the moment here too.

Because the distance measure distances According to the measuring axis.

But this is based on which axis s indicators sets.

Now we are here with Cartesian coordinates I will try.

The little circular coordinates will enter

but the simple circular coordinates examples will suffice.

Or, in the second part of the course the second This skin

As more and more of the notes

wide application and examples circular We will see coordinates.

But here we will make a start the second part of the course

If do not want to take this

an initial circular coordinates would have seen the.

Well a little on one side of the second

will be part of the issues in an ad in the is going on.

Because the circular coordinates is important because Many models in the physical sciences should

statistical requirements in the sciences circular out coordinates are done.

Now let's start with something simple that last a very simple thing, but both

useful in our calculations as well as simple is going to start a integrals.

According to a rectangle's center of gravity moments.

Get a base width b.

Height to get.

As we get to the center by means minus from two to divide divide into two minus b

b divided by two divided by two outgoing boundaries is there.

Areas of very simple integral d.

d the limits of the integral properly We need to leave.

than two years minus b divided by x plus b divided into two

plus two from minus divide divide into two going up.

Of course we can do it in reverse.

d y d x as.

But how do you do that's known field

This turns out to be a b integrals.

Judging from the first and second moments

center of gravity of the moment nothing to do with We know that.

Because we take the center of gravity according to

in advance that they go to zero We know.

D y d y times if we do really well here x again, but now we see the limits fixed

The y integral to the variable gives zero when we separated.

Because more than two years plus b minus b divided section negative values when carrying.

After going to the surplus value or numerical as

If you do the above two squared divided by y value

minus the square value, since there is the following

and it turns out plus minus zero parts.

Similarly, the m y in this section X, of which are zero

the average value of the center of gravity Easily turns out to be zero.

To the moment of inertia, the second When we look at the moment

y is the integral of this time frame.

i x for x.

I have the integral of x squared for y.

Gene boundaries with constant values and for determining

function components can be removed This year, the square function

If a thought crossed y squared x

an integral over a year on the square integration.

As you can see from the integration over x comes.

Here it twice from the year divided by three cubes b divided by the square of the two sounds.

Maybe it elsewhere 've seen.

b cube turns divided by 12.

i y in the same process is done.

However, because different tasks b to a'yl

xin interchangeably tasks y'yl is coming.

For him here instead of a b b a cube cube is coming.

When we look again i x to y symmetry shows the importance.

You can still reserve variables and because it is hard limits

x y is taken as the x by y times it gives each zero.

Therefore i x of y is zero we see.

This is a good symmetry of x and y escape from a scale showing.

But the symmetry is zero if no escape.

I.e. shows that symmetrical both x axis and y-axis according to.

I x squared plus y squared zero of this area on the integral.

I immediately see that x x and I y that we obtain in this manner.

Yet this time the same rectangular edges

in the case where by the axes moments can be calculated.

This is also important in applications, good

As an exercise it as homework I'm leaving.

Something to connect it to the previous one possible but

so at this stage we're not our.

We find examples of integration makes sense for we want.

Random write functions in the integral Instead of trying to be masters.

Integrally simpler but integrals more important to write limits.

Now here's a right triangle

axis perpendicular to the sides of the triangle cul

When the first and second moments We want to find.

From the first moment of the center of gravity coordinates.

Here are a convenience guidance is we do

he's also able to easily check accounts I am writing boundaries.

Here x is held constant over before

y can be taken as an integral constant could be taken at x on hold.

Already in the previous section, many

We have seen examples of the triangle on the as integral.

That's the moment where f x and y according to the nature x or y or x square or

square or x and y where y including As you can see i x and y can not be zero.

Because here too if you want a symmetry You can find

According to the symmetry axis, but it does not exist.

For this reason, it would not zero advance we could find.

In the rectangle on the other side in advance 'd find there would be zero.

A second assignment

but it also differs slightly from the first is no different.

And left the two limit variables limit in both,

a right isosceles triangle limit y axis

over a height of oturtulun The presence of the moment.

Less than a base that goes up to a two a, b is the height at

course of the field work a times b divided by two two from clear that a b.

Here are the y-axis symmetry hence the x-coordinate

in terms of the center of gravity at zero We know that.

One-third of the weight from the base of the triangle

We know that from the height divided into a trilogy.

All of these integrals your çıkarabilme I'm waiting.

This account can be made is not difficult.

We have seen similar regions.

Here you can see immediately that x is held constant y take on the integral is not appropriate.

Because it is composed of two types, then.

You can find two of the integral.

However, the constant hold x on y If you take the integral of a previous example

Likewise, these integrals such as the writable.

Basing do not properly challenge the boundaries and results can be found.

According to the center of an isosceles triangle We can find moments.

It is also interesting in applications and a as math problems

To do the job properly in a nice exercise.

This will show you the way things given.

The geometry of the triangle from the already features that you know.

They also left as an exercise for you I'm waiting.

Now I want to pause here.

We will now, because a new topic.

However, this issue is a huge topic circular

coordinates, but you want to leave this course, carry over

The second part, at least do not want to some circular

See coordinates so that even a short I'm doing a presentation.

Already grasp of the main ideas In solving the problems

would not be a challenge, but related There are interesting applications.

Them in the next lesson, so very variable

function of the number two in the course detailed

Many will see and also serves as a work are examples.