Definite Integrals (part 4) Khan Academy English Turkish

Definite Integrals (part 4)YouTube↗ IA↗(long shot)
Education 217.3K views 466 likes 18 dislikes 22 Oct 2007

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I'm now going to use definite integrals to figure out the

Belirli integralleri kullanarak birkaç eğrinin altındaki

areas under a bunch of curves and, if we have time, maybe

alanları zaman kalırsa da eğrilerin arasındaki

even between some curves.

alanları bulmayı göstereceğim

So let me right down the fundamental theorem

O zaman kalkulusun temel teorisini

I know I covered it really fast in the last presentation.

En son sunumda üstünden hızlı geçmiştim

Just to make sure you understand this formula.

bu formulu anlamanız gerek

The last couple presentations were really to give you an

Son sunumlarda bu formülü size

intuition for this exact formula.

tanıştırdım

Let's say that big f prime of x is equal to f of x, right?

F(x)'in f(x)'e eşit olduğunu söyleyelim, değil mi?

That's also like saying that the -- that's equivalent to

bu ayrıca F(x)'in

saying that f of x -- big f of x -- is equal to the

f(x)in ters türevine

antiderivative of f of x, right?

eşit olduğudur değil mi?

Well, let's just that it's one of the possible antiderivatives

Bunun f(x)'in herhangi bir anti

of f of x, right?

türevi olduğunu düşünelim

Because there's always a constant term here and you're

çünkü her zaman bir sabit terim vardır

not sure whether it is.

ve bunun sabit olup olduğuna emin olamazsınız

And this is why people tend to use this standard because we

Herkes de bu yazılışı kullanır çünkü

know that f of x is the derivative of big f prime of x.

f(x) F'(x)'in ters türevidir

Big f of x is just one of the antiderivatives of f of x.

F(x) f(x)in ters türevlerinden sadece biri

So this is a little bit not true, but I think

pek doğru değil ama bence de

you get the idea.

siz konsepti anladınız

But the fundamental theorem of calculus tells us if this top

Kalkulusun temel teorisi bize eğer tepede

line is true, then the definite integral from a to b of f of x

yazan doğru ise a'dan b'ye f(x) dx'in olan belirli olan integrali

d x is equal to its antiderivative evaluated at b

b'de hesaplanan ters türev eksi a'da hesaplanan ters türevin

minus its antiderivative evaluated at a.

eşit olduğunu söyler

And I know I said here that big f isn't the only

Şöyle ki ben F(x)in sadece ters türev

antiderivative, right?

olduğunu da söylemedim, değil mi?

Because you could at any constant to this and that would

Çünkü ona herhangi bir sabit terim ekleyebilirsiniz ve

also be the antiderivative.

o da anti türev olacaktır

But when you subtract here, the constants will cancel out.

burada çıkarırsanız, sabitler birbirini götürür

So it really doesn't matter which of the

o yüzden hangi sabitleri seçtiğiniz

constants you pick.

The constant actually doesn't matter.

sabit aslında pek önemli değildir

So that's why I actually said the antiderivative.

işte bu yüzden anti türev dedim

But let's apply this.

bunu uygulayalım

You might be confused right now.

kafanız şu an karışmış olabilir

So let me draw a graph.

Look how straight that is.

ne kadar düz olduğuna bakın

Draw the x-axis.

x eksenini çizeyim

Not perfect but it'll do.

mükemmel değil

ama yine de çizeceğim

Let's say that my f of x is equal to x squared plus 1.

f(x)'in x kare artı 1'e eşit diyelim

So f of x looks like this.

f(x) böyle görünecek

bu

1

So it'll start at 1.

bir'de başlayacak

It'll just be a parabola.

Let me see how good I can draw this.

ne kadar iyi çizebildiğime bakayım

I've done worse.

daha kötü çizmiştim

So that's f of x.

It's a parabola. y-intercept at 1.

parabol şeklinde, y eksenini 1'de kesiyor

And let's say I want to figure out the area under the curve --

şu anda eğrinin altındaki ve x ekseni ile

between the curve and, really, the x-axis.

arasında kalan alanı bulmak istiyorum diyelim

Let's say I want to figure out the area between the curve and

eğri ile x=-1 ve x=3

the x-axis from x equals negative 1 to, I don't

arasındaki alanı

know, x equals 3.

bulmak istiyorum diyorum

So this is the area I want to figure out.

burası bulmak istediğim alan

I'm going to shade it in.

So this is the area.

All of this stuff.

I want to figure out this area.

bu alanı bulmak istiyorum

And you could imagine, before you knew calculus, figuring out

Kalkulusu bilmeden bir eğrinin

an area of something with a curve -- it's kind

dahil olduğu alanları bulmak

of top boundary.

It would have been very difficult.

o zaman zor olurdu

But we will now use the fundamental theorem of calculus

Şimdi kalkulusun temel teorisini kullanacağız

and hopefully you have an intuition of why this works and

ve size de umarım neden kullanışlı olduğunu

how the integral is really just a sum of a bunch of little,

integralin aslında tabanları dar olan karelerin toplamı

little, small squares with infinitely small bases.

olduğunu göreceksiniz

But if you watched the last videos, hopefully that

eğer son videoları izlediyseniz

hit the point home.

bunu anlamışsınızdır

But now we'll just mechanically compute because, actually,

Şimdi elimizle hesaplayacağız çünkü

understanding it is a bit harder than just doing it.

anlamak sadece yapmaktan birazcık daha zor

But let's just mechanically compute it.

elimizle hesaplayalım

So we are essentially just going to figure out the

sadece x'in karesi artı 1 dx'in -1'den

integral from minus 1 to 3 of f of x, which is x

squared plus 1 d x.

integralini bulacağız

What's the antiderivative of x squared plus 1?

x kare artı 1'in ters türevi nedir

This just equals the antiderivative.

bu ters türeve eşittir

So it's just x to the third -- we could say 1/3 x to the third

x'in küpünü bulacağız-o zaman 1/3x'in küpü + x veya x'in küpü

or x to the third over 3 -- plus x, right?

bölü üç + x diyebiliriz değil mi?

The derivative of x is 1.

And then we don't have to worry about plus c because we're

+c hakkınd endişlenmemize gerek yok çünkü

going to subtract out the c's.

c'leri birbirinden çıkaracağız

I think you'll get the point.

bence anlayacaksınız

It doesn't matter.

You could pick an arbitrary c right here and it'll

buraya istediğiniz c'yi seçip koyabilirsiniz

just cancel out.

ve c'ler sadece birbirini götürür

And we're going to evaluate that at 3 and negative 1 and

bunu 3'te ve -1'de hesaplayacağız ve

we're going to subtract out big f of negative

F(3)'ten F(-1)'i

1 from big f of 3.

çıkaracağız

This is just the notation they use.

bu kullanılan gösterim şekli

You figure out the antiderivative and

ters türevi buluyorsunuz ve nerede

you say where you're going to evaluate it.

kullanacağınızı gösteriyorsunuz

And then this is equal to -- So if I evaluate 3.

3'te hesaplayacak olursam, evet 3'ün küpü

3 to the third power is what?

27 divided by 3 is 9.

And then 9 plus 3 is 12.

artı 9 12 eder

This is just big f of 3, right?

bu F(3) değil mi?

Because I figured out the end -- This is big f of x.

sonunu buldum, bu F(x)

You can kind of view this as big f of x.

bunu büyük f(x) diye düşünebilirsiniz

But not to be confused with small, cursive f of x.

f(x) ile karıştırmayın

This is big f of x.

So this big f of 3.

And then, from that we'll subtract big f

F(3)'ten F(-1)i çıkaracağız

of negative 1, right?

Minus big f of negative 1.

And if we put minus 1 here.

buraya da -1'i koyacağız

Let's see, minus 1 to the third power is minus 1.

-1'in küpü -1'dir

So it's minus 1/3 and then plus minus 1, right?

-1/3 artı artı -1 değil mi

So minus 1/3 plus minus 1.

I think that equals minus 4/3, correct?

bu da -4/3'e eşit oluyor

Maybe I'm making a mistake with negative signs.

eksi işaretleriyle bir hata yapıyor olabilirim

Minus 1/3 minus 4/3.

And I'm going to subtract that, right?

bunu çıkaracağım değil mi?

So if I'm subtracting minus 4/3, it's the same thing

eğer -4/3'ü çıkaracaksam bu 4/3'ü eklemeye

as adding minus 4/3.

And then we have our answer.

cevabımız belli

Actually, it's 12 and 4/3 -- whatever -- units.

12 ve 4/3-birimsiz

12 and 4/3 squared units.

12 ve 4/3 kare birim

We could write this as a mixed number as well.

bunları karma sayı olarak da yazabiliriz

Let's do another one.

şimdi farklı birini yapalım

I'll do a slight variation.

Let me draw again.

tekrar çizeyim

Some coordinates.

bazı koordinatlar

I don't know if I'm going to have time to do it in

bu videoda bunu yapabilcek zamanım kalacak

this video but I'll try.

mı bilmiyorum ama deneyeceğim

Let's say I have f of x is equal to the square root of x.

f(x) in x'in köküne eşit olduğunu düşünelim

So it looks something like this.

That's actually a pretty nice looking, kind of sideways

parabola, I think.

bu yanlamasına bir parabol

This is f of x.

And let's say I have another function.

farklı bir fonksyionumuz daha olsun

g of x which equals x squared.

g(x) de x'in karesine eşit olsun

So g of x is actually going to look something like this.

g(x) de bu şekilde

I was doing well and then something happened.

iyi yapıyordum ve bir şey oldu

And, of course, it'll continue on this side as well.

tabii ki bu tarafta da devam edecek

Because it is defined for negative numbers.

çünkü negatif sayılar için belirlenmiş

But anyway, my question to you, or my question to myself,

neyse benim size veya kendime sorum şu olacak:

really, is what is the area between the curves

eğriler arasındaki alan kesiştikleri

where they intersect?

What is this area?

Well, the first thing you have to figure out is just what

önce bulmanız gereken şey sınır noktalarının

are the boundary points?

What is this point?

bu nokta nedir?

And what is this point?

Well this point, I think, is pretty clear.

bence bu nokta olduğu açıkça belli

It's 0, 0, right?

0,0 noktası değil mi?

They both intersect at 0, 0.

ikisi de 0,0 noktasında kesişiyorlar

And even this point, you could probably do it from intuition.

tahminden de yapabilirsiniz

But if you don't, I guess, want to do it through intuition, you

yapmazsanız da ben tahminle yapmak istiyorum

could just set these 2 equations equal to

yine de 2 denklemi birbirine eşitleyebilirsiniz

each other, right?

You could say x squared is equal to the square

x'in karesi x'in kareköküne eşit

root of x, right?

diyebilirsiniz değil mi?

And then you could do a bunch of things.

birçok şey yapabilirsiniz

You could square both sides or -- well, actually, this is the

iki tarafın da karesini alabilirsiniz,

same thing as doing it by intuition.

bu da tahminle yapmakla aynı şey zaten

But I think it's pretty obvious that the only places where x

bence x'in karesinin x'in kareköküne eşit olduğu

squared is equal to the square root of x are the points x

yerlerin x=0 ve x=1 olduğunu

equals 0, which you already know, and x equals 1.

So this is the point 1, 1.

bu 1,1 noktası

Which is true for both of them.

ikisi için de doğru

And this is more algebra, so I won't go into that

bu daha çok matematik o yüzden pek de detayına

in too much detail.

I'm kind of running out of time.

zamanım bitmek üzere

So we want to figure out the area between these 2 curves.

eğriler arasındaki alanı bulmak istiyoruz

So what we can do is -- maybe you want to pause it and think

O zaman yapmamız gereken-

about it yourself -- we can figure out the area

belki videoyu durdurup kendiniz düşünmek istersiniz-

under the grey curve.

gri eğrinin altındaki alanı bulmak

We could figure out this area.

So we want to figure out -- this is a boundary, right?

burası sınır değil mi

Between 0 and 1.

0 ve 1 arasında

We could figure out this area and then we could figure out

bu alanı buluruz, bulduktan sonra da

the entire area under the green curve separately and then we

yeşil alanı ayrı hesaplarız daha sonra da

could subtract the difference.

ikisinin arasındaki farkı alırız

Which is exactly how we're going to do it in the next

bunu da bir sonraki videoda yapacağım çünkü

video because I have run out of time.

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