Hi there.

Prior to that, in space

curved surface area accounts We have developed methods to.

And a repeat here If we make a compilation of six

have found that the calculation of ways.

Basic and general approach

With this we show d s mockery of the surface in space,

is the area projected onto the plane xy

If the unit of surface area with the z axis perpendicular vector

the interior of the unit vector in the direction perpendicular by multiplying this formula is obtained.

In many problems directly n are available and that account

The formula ever developed a need for would be able to do without.

But the subsequent formulas are improved formula.

It is basic, given the overall result

According to the type of function by calculating able to develop.

An outdoor function of the first kind when given,

The function described surface, n is required here when he calculated

is that of s curve in space surface in the x-y plane

integral in terms of the projection of d We found that it was.

Similarly, a closed surface If defined with the function here again

by x, y and z as function partial derivatives can be calculated.

In the third type of surface with a vector function

u and v are two parameters, but from the parametric representation

d s is the projection in space

denominated in the area of ??the x-y plane We have seen that such a relationship.

Here again, the length j Jacobian emerges.

The two binary j x i of Jacobins,

the row i and

milk threw the remaining two binary matrix determinant.

This is a Jacobian.

je l'first year again when we account After I remove the line and the second column

the remaining two binary determinants j gives the determinant of the matrix y.

Similarly, j k.

j s get it here The length of the vectors.

These are more fully removing the saw.

So far, all we know in the plane

by integrals, but in the plane of a two-storey

When we want to find the area of this terms was always going ahead.

When a curved surface, but this terms that reflect curvature

we choose the function representation by type is calculated in this way.

A new method, the surface again

the parametric representation and you have to calculate in terms directly.

This is a curvilinear coordinates u and v are.

V held constant when x i.e. only one of the vector u

becomes a function of the parameters.

This is kind of a vector function We are aware that the line in space.

Similarly ui sabitleyin de x

v is a function Another line of family forms.

And in this case curvature surfaces in space

d u d v not only their Cartesian coordinates would be if you had a j s.

In this way an extra term for these comes to the calculation of the integral.

A special type of a surface In the area of ??surface of revolution

We have also developed the formula for the calculation.

z in the vertical plane as a function of r

If to such a representation, geometrical surface of revolution

about the z axis of this line a surface obtained by rotating.

Now their various We will see examples.

While our sample following a well we will do the calculations in detail.

d s is the appropriate solution here

d s is the surface area is calculated as the integral.

But usually this kind x y plane of the surface

from the center of gravity length is also important.

According to a z axis, The second moment is also important.

They also will do.

When it calculates the first torque with z d s h divided by slamming it,

this surface along the z-axis gives the center of gravity.

Now let's move on to concrete examples.

The surface of a hemisphere We want to examine

and the scope thereof six different We want to calculate the road.

First, why hemisphere?

Because of this sphere projection by the time we get to the x-y plane,

a circle in the upper hemisphere, will be projected onto a circle,

The projection of the southern half Also the projection sphere,

thus take twice I need to find a full sphere.

Therefore hemispherically We are working on the

come on top of the projection lest occurs twice.

Multiplying the results obtained by two We all find the area of ??the sphere.

We know that already.

Do you know that since time immemorial.

Since the dawn of high school as well.

The surface area of a sphere We know that four pi squared.

When a radius.

Now let's start the general notation.

We know that such a formula.

A half sphere equation representation of the sphere diameter x

squared plus y squared plus z is equal to the square of a square.

In this formula, n we need.

We need to know this n.

n is the length of this surface gradient We know that division.

But the steep gradient vector gives a length not necessarily.

A length that obtained here to provide

The gradient vector we divide longitudinally.

Very easy gradient, derivatives with respect to x two x's.

derivative with respect to y two years.

two z z derivatives.

Because every time the other variables for doing the hard tasks.

When you calculate the length of this four x squared plus y squared four four z

be the square root of the square.

Now comes a simplification immediately.

Because we know that,

sphere from the equation, x squared plus y squared plus z squared, is equal to a square.

Therefore, here we put a square Remove the square root of a.

Also here are a four.

This is the time when the square root of two and would simplify the above two.

Therefore n x y z position becomes a division of the vector.

Unit size that we can control.

You are calculating the length of x squared plus y z squared plus the square root of the square will be.

He therefore squared equation, out of here.

So this vector we k in the z direction, ie with

we take the inner product of the unit vector, k is zero is zero.

Therefore, zero times x plus zero times the sum of z to y once.

Here the denominator remains the only z.

There are already a denominator.

He already will not be affected by this process.

Thus, on the surface of the sphere d s infinite

There are small areas in this way.

d a d x d y n k'yl internal multiplied by z, we have split.

This is of course because it is in the denominator of this mold a share of our time goes in the denominator.

Such an integration.

Should we write it in Cartesian coordinates z

rather than a squared minus x squared We will write minus y squared.

As you can imagine this sphere, x axis of the hemisphere,

x y axis on the x y plane of will occur in the projection of a circle.

For this, the appropriate Cartesian coordinates not coordinate circular coordinates.

R d r y for him d.times.d d theta are writing.

x squared plus y squared de We know the r squared.

Then the circular coordinates d s in this way is it more simple.

Now there is the issue of this integral account.

Usually, such that the square root We are trying to get rid of the square root of time.

Therefore, a squared minus Let's say u r squared.

d r we need.

Minus two times r d r d u will.

Already here we have r d r.

Therefore, our work will go quite smoothly.

See here for lack of theta, integral removable suit.

Reset the integration of two theta pi a full circle because we are wandering.

the relevant parts of r'yl d r r r squared minus a squared.

You can now do the integral We find the value on theta.

Here there was an a.

Here comes a two-pin.

Two pin a.

This term r squared minus a squared minus

As first force and if we define r squared minus

we call the square, this u minus one-half second would force.

If r d r d u would have a negative divided by two times.

This integral immediately the integration can be done.

Limit one should pay attention to the course.

A square is happening is zero.

While RA is happening is zero.

This integration is very simple, defined integration.

Get minus one-half of the integral

Do you know a We are increasing exponentially force.

When you add a trailing minus a half a split in two and one-half we divide into two.

Divided by a slash of course, goes to the two share.

Here we had the minus sign.

This dilemma was to simplify each other.

There are minus pi.

Now it at zero when calculating a square when he calculated at zero,

here comes two of because it is the square root of u.

Because it comes with a minus sign This also improves the negative.

We find two pi squared.

Because it results in a known All of the surface area of the sphere

According to a squared four pi is half will be two pi squared.

As you can see no formulas without memorizing only the perpendicular vector

knowing that from the gradient able to take account of the convenience.

Here again, a common meeting integral type, have a square root.

When the square root generally see the transformation of our business.

In the second method implicit function Let the representation.

The open function, but also have representation As you can see with the function off

sphere equation easier.

Gradient are account.

These f x, f y, f z.

We take it for putting in the formula instead.

That's our formula, d s is the formula:f x's square, the square of f y, f z squared.

You can see them when we put Nothing like the previous situation we encounter.

Gene x squared plus y squared We are also a plus z squared.

Comes out of the square root We find here a.

Take the square root of four will be two.

This dilemma also in the denominator by two sadeleÅip found previously divide z

d times to come.

Of course, the next integral is the same as the previous one.

Because d s the same.

And this circular coordinates We did turning.

We're doing the same thing here.

If we make representations with clear functions, See there the formula

were as follows:x is a plus for There squared plus the square of f y.

You have the square root.

Now get some more of this integral As challenging but not so difficult.

See derivative chain rule,

Every time again how we see that useful.

A squared minus x squared minus Do u say y square,

that one half of the second force, one-half of its derivatives will fall forward.

over one-half of a will raise will be divided by two minus one.

D u d * will be also divided.

He divides d u d x is a minus two Give the equation for x,

that these partial derivatives can be customized.

fy is done in exactly the same way.

x instead of y.

Our formula is a plus for us x squared plus y squared had fun.

F x squared plus x squared denominator for a

Because the same common y'yl We can now combine the denominator.

A squared minus x squared minus y squared.

y frames are also here.

Now when you bring them to a common denominator See squared minus x squared minus y where a

There is a square.

There is unredeemable plus x squared plus y squared.

x and y take each other, remains only a squared denominator.

There is nothing in the denominator is already changing.

Was the common denominator.

You take the square root of d a'yl We find d s gets hit.

Here are the previous two is here s is the same expression is reached.

Therefore integral to again, you do not need anymore.

Because I've done it integral.

Altogether this integration.

Also found here are merely d s.

Now the parametric representation brings a new dimension.

A new type of solution method.

We have to account for it, this j.

j u and v by a partial was the vector product of the derivatives.

Now of course, the surface of the sphere I need to know the parametric equations.

a radius of the sphere such a parametric representation.

If you say u fi, theta, if you say you have.

And v is the request here more concretely,

The following is happening and theta functions.

So fi fi varies along the curve We're going through the meridians.

Tete geographical terms the way again If we use the latitude you're going through.

Our previous coordinates Judging see here,

When replacing the plug Open're going through.

When replacing the theta We're going out latitude and

one way of them an area is achieved.

But of course every time they You do not need to go back and do.

The benefit of this formula.

Now this by x the plug If we take the derivative is easy.

From sine cosine will be.

Others are the same.

Here again it will be the cosine of the sinus.

From here the minus sine cosine will be.

V, i.e. by theta When we received the derivative, the

See where a sinus fi common to both.

Minus sine cosine come from.

From sine cosine will be.

In the last term, the last component, In our theta no.

Therefore, the derivative with respect to theta is zero.

Vector multiplication means the first line will write i j k.

The second line will write the first vector.

The third line will write the second vector and we will calculate the determinant.

Since there are i j k course will be a vector.

See here have a partner.

We take this outside When there is no content.

Here, too, has an a.

We also take out When we create a square.

This i the first to find components We are temporarily closing line.

Vertical are closing.

Here zero plus sine-squared fi cosine theta involved.

Similarly, the term j'l are account.

Similarly, the term K in composition are account.

Here in this vector, we find the vector j.

This gene of the vector length When you calculate what

You will see that much simplification.

Already it Jacobins We saw the same sample.

Because we calculate the Jacobian from the also nothing more.

Now where e, j's easier to find a frame.

Then take the square root.

it is a four squared.

See it on the square had four sinus.

There cosine square.

Here are four sinuses.

There cosine square.

Sine, see here are the sine-squared.

Therefore, the common sine-squared as going out.

Inside the sinus remains bi sine-squared than four due to its square.

But this is a time cosine in a time frame, sinus

multiplied to the square, meeting.

As you can see here a bi occurs.

See here that when there is a once again a very interesting structure b.

KosÃ¼nÃ¼s square sine-squared fi fi is happening.

He is also a giving.

So the inside of the brackets completely.

These are not coincidences.

Characterized in that the global koordinatar, nature.

Therefore important is happening already.

As you can see how much was simplified.

When we take the square root of it Of course, we find a squared sine fi.

Here when we calculate d s,

divided by the squared sine f j z.

z j from here we know what it was.

This is a squared sine cosine fi fi.

The following are the more simple sine see here.

I sadeleÅtirel one of a square.

Keep a a a a below above.

Because I see a time where the cosine fi in the z coordinates are gone global.

X of the first component, the second component y z the third component.

This simple geometry We found with the projection.

If you do not remember the formula is given.

Here the denominator means that a occurs.

In the denominator of the z ouÅuy.

So this is already one, two and three the same that we find our approach.

Thus be of integral will be the same as it should be.

Now here you have an interesting deÄiÅikik.

The following respects u and v coordinates directly in terms

possible to calculate the integral.

See the previous page j s were calculated.

j s, has a squared sine fi'y.

You do not need to recalculate.

By doing this these accounts can be found, of course.

Integral j h

V. Di Di was once the general formula.

Although means V d, and d d theta d f can say.

As you can see again the integral variables are separated.

Because it does not function at all in theta.

Therefore teta'l sections are separated.

going out for a frame to be fixed.

Here on d theta integral b will give us two p.

Backward integration is to plug the sinus fi d.

Let us also note the limits.

We are working half the sphere.

Theta hemisphere from the Arctic, sorry f is equal to the equator are coming from scratch.

There, a ninety-degree fold would have.

So it's going pi divided by two.

The following is the sine integral.

This is also very easy to EUR minus cosine fi.

The above limit, below the limit value.

See here for two pi squared occurred.

This is the first integral and at the beginning of coefficients.

This cosine of the above fi'y value zero in pi divided by two.

Remember Flat trigonometric circle.

Angle is zero at the North Pole.

Cosine of the angle is zero.

Fine, f is equal to zero when you put the cosine becomes zero.

But there is a minus sign.

Therefore, these two minus one another taking this integral remains zero.

Here are the two pi squared.

It should already gene We find the same result as.

This approach is completely different.

In all of ÃbÃ¼rki curvilinear hemispherical surface so the x and y

We were getting to the plane of projection and views.

A circle was happening there.

He's on the circle in the plane used to calculate integrals.

Here it directly fi out coordinates and theta

in space on the surface We calculate the integral.

And a supremely simple integral output.

No conversion is needed.

This is because the global spherical coordinates

surface of the natural is the fact that coordinates.

So this curvilinear coordinates such benefits also need to remember.

Surfaces of revolution, we know that.

But if you receive a half-circle, I'm sorry if you receive a quarter of a circle,

Such a half-axis in a quarter circle.

When you rotate it here the surface of the hemisphere is formed.

Thus, the function z is equal here A squared minus x squared minus y squared,

from the equation of the sphere.

But for x squared plus y squared is r squared,

z is equal to the function f r We find here.

According to this variant should run.

Here you force a split second he approaching easier to calculate.

Here bi with a minus sign one-half times together.

Insiders derivative is minus two.

Therefore twos and in the denominator goes here

minus a split second force is going.

One plus for the square of the base We find it again when you get in

we have encountered before and completely integrals we encounter.

This gives a known result.

But see here on theta integral surface of revolution is beginning to

and this formula was taken from the inside lies in the effect of theta, theta contribution.

For him it is a surface of revolution Because a convenient time

automatically that an integral You can reserve in formula.

Contents of this formula that includes theta, contains.

Now I want to give some homework.

Follow the same steps, how some of the course immediately

You can see that, but I would advise you to do.

weight, based on x-y plane We want to find the center.

Our finding that the first moment z'yl have obtained multiplying.

In this area divided by the area We have calculated that pi squared.

It's the center of gravity.

You to account for this integration I want it to six different way.

Bi in the spherical surface of this technology,

naturally in the study of many events something that may be encountered.

A machine element of a global consider physical,

substantially or an electromagnetic space on a sphere

consider the distribution or A heat conduction through a sphere of

with the distribution of the heat flow passing here You can deal with such problems.

And here it

The second moment would need.

The second moment of x squared plus y squared.

r squared x squared plus y squared anyway.

This also accounts for this I'm waiting for you to do.

As results given in You can check your answers.

Yet there's an assignment.

A previous version of this a little more special.

We are one of the hemisphere We found the surface area.

Hemisphere rather than a cone of here ALSA or something like the truth

to take away the arc of a circle Even if you turn about the z axis

not a half-dome, which hemisphere but not a dome for zero clear,

making an angle of, for zero angle with the z-axis less than a half-dome that you can find.

More generally becomes a problem.

Here is a side view.

Eliminate them if you receive a section, these cone

to two, the manufacturer is going on.

A dome that

circle arc of a circle to be obtained.

What is the difference from a previous problem?

There are very few differences.

In spherical coordinates I want you to do this.

Because you can not do in the other coordinates.

You can, perhaps, but supremely You can meet with integral harder.

It is not very difficult but more of a integrals must be tall.

Hemisphere for zero angle pi divided into two parts was coming up.

However, one for zero When we come to terms,

Open for zero for zero comes up.

sphere but has a yarÄ±Ã§apÄ±y

If we say is, x and y are in the plane.

Circular coordinates are in the distance.

This is the sine of a fi who is possessed is reset.

Current distance.

This lower rim of the dome ring

This is not the semi-dome r would be too small.

Reset is for sinus.

These guidance as The information given to you.

While this area accounts the limits of integration from scratch

You will receive fi to zero.

In the previous problem from scratch had received up to two pi divided.

Similarly z'yl hit d s, the

integral from zero again f you will get zero.

rar hit square again from scratch You will receive fi to zero.

Therefore, previous equiv problem completely,

equivalent in these results we can say f to zero if pi divided by two,

See for pi divided by two is zero if we do Our hemisphere comes dome.

Efe, the cosine of 90degrees is zero for the cosine fi

becomes zero and back to zero be two pi squared.

E also know that half of the surface area of the sphere.

Similarly wherein kosÃ¼nÃ¼s for zero fall.

The cosine reset for fall.

There are such a control.

There is a supply of the compound of formula.

Gene homework.

A cone area and moments I want to find.

Where b is a cone surface of revolution.

But it's the way all kinds of I want you to account.

Because that's our goal a little not very random surfaces,

integral, not too messy, and simple integration with

significantly on surfaces improve our skills.

This cone, cone, since elementary school, the maybe some of you might have seen.

If you cut to the middle, If you turn in a plane such

The surface of the cone ring segment means.

Height h, a is the radius of the base

The length of time the generator side square is the square root of a squared plus h.

For this, the problem here is While you will need the following.

Need cone equation.

This cone of the equation x squared plus y squared minus the a and h are

when given a squared minus h z square or a square times

circular coordinates If you see translations

where r squared x squared plus y squared where z has to be square.

Everywhere that squares Take the square root of z

is equal to the right where the interests of the equation.

Divided by the height h of the slope of this line.

See consistent formula here As we come to this conclusion.

Here in the software For convenience,

plus h squared divided by a square, If the square root of alpha

surface and torque and weight center involved in this way.

One more example.

Let's take a paraboloid surface.

Dimensions along the its length to come out I have a one-half to divide.

Here are the x squared plus y squared.

z equals three divided DOUBBLE roots, in the times of

surface with a plane that Let's cut the surface of the paraboloid.

As the situation is going.

There such a Paraboloid.

This even as a surface of revolution a Paraboloid obtained.

If you say x squared plus y squared r square There are a function of z is equal to r squared.

This paraboloid surface different I want to find ways.

N find a single thing to be aware of.

E n very easy to find.

Which kind of representation of functions If you receive be easy.

Moments already given here.

They see it as the moment If you do not want, with physics

If you do not want to take care of, see them as an integral.

ÃÃ§neml find the d s.

The steps in the same sphere We have solved a very detailed here,

If you follow the answer to this You can also find.

Now here I gave the details.

Once you get back before hesaplasak as a surface,

z is a function of r as x squared plus y squared had.

If it is not square, we find this formula.

Upon receiving derivatives here by r, As you can see the dilemma goes.

is a divided leaves.

Here we put them This formula is a plus

f is the square root of the square base d r'yl integrated'll hit.

Why had received three roots?

Because the root of three is taken A scratch is going to.

The root cause of these three.