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Lecture Comments (2)

1 answer

Last reply by: Professor Hovasapian
Thu Dec 17, 2015 12:59 AM

Post by Gautham Padmakumar on December 12, 2015

42:00 did you forget to put the differential of x in the inequality

The Definite Integral

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • The Definite Integral 0:08
    • Definition to Find the Area of a Curve
    • Definition of the Definite Integral
    • Symbol for Definite Integral
    • Regions Below the x-axis
    • Associating Definite Integral to a Function
    • Integrable Function
  • Evaluating the Definite Integral 29:26
    • Evaluating the Definite Integral
  • Properties of the Definite Integral 35:24
    • Properties of the Definite Integral

Transcription: The Definite Integral

Hello, welcome back to www.educator.com, and welcome back to AP Calculus.0000

Today, we are going to talk about the definite integral, very important.0004

In the last lesson, we used the following definition to find the area under the curve.0010

Let me see, what should I do today?0015

In the last lesson, we use the following definition to find the area under a curve, f(x).0024

f(x) is the function, the curve of the function but we just say curve f(x), that is fine, on the interval ab.0055

We said that the area was equal to the limit as n goes to infinity.0065

N is the number of approximating rectangles, we take the rectangles thinner and thinner,0070

of the f(x sub 1) Δx, + f(x sub 2) to Δx, so on and so forth, f(x sub n) Δx.0076

If we have 15 rectangles, we have 15 terms in the sum.0095

Δx was the width of the particular rectangle, we just added them up.0101

I will go ahead and express this as a = the limit as n goes to infinity.0109

We have the summation notation, the shorthand for this.0114

When we picked some index i, which runs from 1 to n, whatever that happens to be.0119

F(x sub i) × Δx, here the Δx is equal to the b – a/ n.0125

It is the right endpoint of our domain, the left endpoint of our domain of our interval divided by the number of rectangles that we wanted.0137

x sub i is actually in the little sub interval x sub i - 1 x sub i.0148

All this means is that when we break up the interval, let us say we have x sub 3, and we have x sub 4, 0160

something like that, and we have a over here, b over here.0170

In this interval, our x sub i is somewhere in here.0174

Sometimes, we pick the left endpoint, sometimes we pick the right endpoint.0179

Sometimes, we pick the midpoint, and that it can also be any other point in between.0183

That is all that that means, this part right here.0187

Let me go ahead and actually draw a little picture of that.0192

We have our axis, we had a curve, we had a, we had b.0198

A is the one that we called x0.0205

Let us say we had 1, 2, 3, 4, this would be our x sub 1, this would be our x sub 2, x sub 3, x sub 4,0208

and our x sub 5 is going to be our b.0221

This just means somewhere in there.0226

Our x sub 4 is going to be somewhere in that little sub interval, that is all that means.0230

We are going to define the definite integral.0241

We now define the definite integral.0251

I will go ahead and put this in red.0263

You know what, I think I want to use red.0273

I think to prefer to just go back to black, how is that?0278

We will say let f(x) be defined on the closed interval ab, divide ab into n sub intervals.0284

Everything is exactly as we did before, each of length Δx which is equal to b – a/ n.0311

We will let x sub 0 = a, then we have x1, x2, and so forth.0328

We will let x sub n = b.0338

The x sub 0 is the left endpoint, the x sub n is the right endpoint.0344

Then, the integral a to b, f(x) dx, the definite integral of the function f from a to b is equal to the limit that we had.0349

It is equal to the limit as n goes to infinity of the sum i = 1 to n of f(x sub i) Δx.0369

Again, where x sub i is in the sub interval, x sub i – 1, or the x sub i is somewhere in the interval x sub i - 1 x sub i.0384

In other words, x sub 3 is going to be somewhere between x sub 2 and x sub 3.0411

That is all that means.0420

If this limit exists, if the limit on the right exists, in other words,0422

we go through the process of finding the Δx which is the b – a/ n, we find the f(x sub i) based on the x sub i.0438

We form this thing, we form the sum of this thing, and then what we get is going to be some function of n, and then we take n to infinity.0446

If we end up getting a finite limit, if the limit exists, we say that f(x) is integrable.0454

This, the limit that we get is that definite integral of f(x).0474

If not, if the limit does not exist, then not.0485

If not, then not integrable.0489

This is the definition of the definite integral.0496

We started up with an area, we use this limit to find an area.0500

Now we are actually using this definition of area.0506

No longer is area, we are actually defining it as something that has to do with the function itself.0511

Something called the integral of that function.0519

This thing, a to b of f(x) dx is a symbol for the entire process that we go through, 0527

entire process of summation + the limit as n goes to infinity.0548

This is a symbol for the entire process of forming the function, taking the sum of the function, then taking the limit.0564

Let us label a couple of things here, say what they are.0573

This thing, the f(x), this is called the integrand.0581

This right here is called the lower limit of the integral.0593

This right here, analogously, is called the upper limit of the integral.0599

When we read this, we always say the integral of f(x) from a to b, not from b to a, from bottom to top.0606

This thing, this reminds us of the independent variable.0615

If the independent variable of the function is x, this has to be dx.0628

If the independent variable of the function is r, this has to be dr.0632

Reminds us of the independent variable and is the differential version of the Δx.0636

This symbol, it is in the shape of an elongated s to remind us 0675

that integration is just a long summation problem.0697

As it turns out in mathematics where the only thing that you can never ever do is add two numbers,0721

or 3 numbers, or in the case of calculus an infinite number of numbers.0725

That is what this says, this is reminding us that all we are doing,0730

when we are taking the definite integral of something is we are forming a big sum, a hundred numbers, 2000 numbers.0734

Or when we pass to the integral from the limit, when we take the limit and we passed to something else, in this case, the integral an infinite sum.0743

That is what is there to do, it is their to remind us that all we are doing is we are adding a lot of numbers together.0751

The definite integral is a number, if it exists.0761

If f(x), if the function that we happen to be dealing with is greater than 0 for all x in the particular interval ab,0784

in other words, greater than 0, it means above the x axis.0798

Then, this number just happens, that is the important thing, it just happens to be the area under a curve for f(x).0804

We started off by dealing with curves and we use this limit that we defined, the limits of the sum of such and such.0829

It turns out this integral is the deeper concept, the integral is a number.0838

It is a number that is associated with a function over a certain integral.0845

We define that number by going through this process of doing the limit.0849

It so happens that, if the function is greater than 0, the area under the curve happens to equal the integral.0853

We are not defining the integral in terms of area.0863

What we are actually doing is we introduced area first 0866

to sort of give some physical conceptual motivation, for this thing that we are trying to do.0868

As opposed to just introducing that this is the integral.0875

Here, you form the sum, you take this limit and that is that.0877

That is very abstract, we start with the area.0881

We are defining this, we use that limit to define the integral.0884

It is the integral that is the deeper concept.0889

It just so happens that it happens to match the area under the curve.0894

It is the integral that is the deeper concept.0898

It can stand alone, it does not need to be associated with area.0900

As a mathematical tool that happens to give us the area, that is what is happening.0907

Let us talk about that.0915

We just said that if f(x) is bigger than 0 happens to be equal to the area.0925

But if we have something like this, let us go ahead and go this way.0930

Something like that, let us say this is a and let us say this is b.0939

Let us just call this a’ and let us call this point b’.0953

For regions from a to a’, the graph of the function is above the axis.0964

If I take the integral from here to here, I’m going to get an area.0973

The same thing here, from b’ to b, the function is above the x axis, it is positive.0977

Therefore, if I take the integral, if I take that limit, I’m going to get a and area.0983

What happens though if the function is actually below?0989

Here, the area is positive, here, the area is positive.0992

The approximating rectangles are just, you are taking some Δx and you are essentially taking some f(x).0997

F(x) is negative, what you are going to end up with is a negative area.1007

For the regions below the x axis, the area, I will go ahead and put it in quotes, is negative.1013

Because this f(x sub i) dx, because these f(x sub i), these are negative.1039

They are below the x axis.1053

F(x) here is down here, -16, -14, -307, whatever it is.1054

These are negative.1061

When you take the integral from a to b f(x) dx from here, all the way to here, which involves this and this, this is the definite integral.1070

But it does not give you the area under the curve, it is a number.1091

What it does is it gives you sort of a net area.1096

It is a net area, if you will.1102

Let us say this area here was 10 and let us say this area under here was 30.1107

Let us say this area under here was -50.1113

The integral from a to b of this function is going to be 10 - 50 + 30.1121

I mean, technically, we can go ahead and take the absolute value, if we were asked what is the area under the curve?1131

We could take 10 + the absolute value is -50 which is 50, 60, and 70, 80, 90.1138

Yes, the total area, in terms of actual physical area is 90 but the integral is not 90.1144

The integral is 10 - 50 which is -40 + 30 – 10.1152

The integral is a number, it can be associated with an area if the function as above the x axis.1158

But it does not have to be associated with an area.1164

It just so happens to be the same sort of tool that we can use to do so.1166

Integral, separate mathematical object altogether, has nothing to do with area.1170

That is the important part.1177

The definite integral is a deeper concept, I should say it is the deeper concept.1184

It is a number which we can associate with a given function on a specified interval.1208

In other words, if I gave you some f(x) and if I gave you some interval ab,1236

with that function on that interval, I can associate some number.1246

In other words, I can map a function on an interval to a number.1254

That is what this is, mathematically, that is what is happening here.1259

Given a function and the interval, I can do something to it.1264

In other words, I can integrate it.1267

What I get back, what I spit out, after I turn my crank and do whatever it is that I'm doing to this function, I get some number.1271

For functions greater than or equal to 0, we can then associate area with this number, the integral.1281

The integral comes first, the number comes first, the area afterward.1309

We just introduced it in a reverse way.1317

Note the definite integral, if integral of a function, I guess the real morale that I’m trying to push here,1328

the definite integral of a function does not have to have a physical association.1345

The definite integral exists as a mathematical object.1367

It exists as a mathematical object, independent of physical reality.1382

The mathematics that we used to describe the physical world, it is not there to describe the physical world.1396

The mathematics exists, it is a tool.1405

Whether we discovered that we can associate this math, 1410

we can use it to describe the physical world but it does not come from the physical world.1414

In other words, the existence of the physical world does not give birth to mathematics.1421

Mathematics exist independently as mathematical objects.1426

It just so happens that physical reality happens to fall in line with mathematical description.1431

That is what is going.1436

The integral exists, it happens to be associated with an area, if we needed to.1439

As we will see later on in future lessons, it is going to be associated with more than just area.1443

It is going to be associated with volumes and all kinds of things, which is why calculus is so incredibly powerful.1448

Let us say a little bit more.1461

We chose Δx = b – a/ n, in other words, I should that is, we made Δx the same width for every integral.1464

Δx the same for every approximating rectangle.1489

They do not have to be equal, they do not have to be the same.1510

Always approximating rectangles, we made them that way, of uniform width,1522

simply to make our lives easier so that we can actually do some work, do some math.1526

They do not have to be the same, each one can be different.1533

You actually end up getting the same answer.1535

They do not have to be the same, it simply makes things easier to handle.1539

Yes, that is the only reason.1543

It simply makes things easier to handle.1546

Also, we said that x sub i is in x sub i -1 to x sub i.1560

By choosing right endpoints, in other words by choosing x sub i as the right endpoint of that particular sub interval, that is x sub i = x sub i,1580

again, we just make things easier and we make things more uniform, more consistent.1611

It is not so haphazard, not without just taking random points.1622

We are being nice and systematic.1625

In any sub interval, we take the right endpoint.1626

That is it, nice and systematic, keep things orderly.1629

Again, we just make things easier.1630

Or I should say, not every function is integrable.1640

Just because a function exists, it does not mean that we can actually integrate it.1644

In other words, when we run through the process of finding it, taking the summation, taking the limit, the limit may not exist.1650

Not every function is integrable.1656

Our theorem, if f(x) is continuous on the closed interval ab, if it is continuous,1667

or if f(x) contains at most a finite number of discontinuities, 1698

the operative word being finite, then f(x) is integrable.1723

We have a criteria now to decide whether a function is integrable or not.1734

If the function is continuous on its domain, on its domain of definition, on its interval that we have chosen.1739

If it is continuous, it is integrable, I can integrate it.1744

I can find the number associated with it.1749

Or if that function on its sub interval, the ab has a finite number of discontinuities, I can integrate it, it is integrable.1750

For now, we will evaluate the integrals, the definite integrals.1768

We will evaluate the definite integral using the definition.1787

In other words, we will use the summation limit.1798

We will use the definition to evaluate these things.1799

It is going to be a little tedious, but that is fine.1802

We want to get some sense of what is actually involved here, using the definition.1805

Later of course, the same way that we did with differentiation.1813

Remember, we have the differentiation, we formed the quotient f(x) + h – f(x) divided by h.1816

We simplified that, then we took the limit, that gave us the derivative.1822

That is the definition of the derivative.1825

Then, we came up with my simple quick ways of finding the derivative.1827

We will do the same here.1830

We are going to use the definition first and get our feet wet with that, just to get a sense of what is going on.1831

And then, we are going to come up with quick ways of doing the integration.1836

You have actually already done some of them, antiderivatives.1840

An antiderivative is the integral, that is what is going to be happening.1843

For now, we will evaluate the definite integral using the definition.1848

As such, we may need the following to help us.1853

We may need the following properties to help us.1861

If I have a sum i goes from 1 to n of i, there is a closed form expression for that.1878

It is going to be n × n + 1/ 2.1886

In other words, what this is saying is that, if I take, let us say n is 15.1890

If I say, out of the numbers from 1 to 15, 1 + 2 + 3 + 4 + 5 + 6 . . . + 13 + 14 + 15,1894

if I add all of those up, I can actually just take 15 × 16 and divide by 2.1902

This gives me a closed form expression for the sum, that is all this is.1908

Just some formulas that are going to help us along.1914

I have another one, the sum as i goes from 1 to n of i², that also gives us a closed form expression.1921

+ 1 to 1 + 1/ 6.1930

I happen to have a closed form expression for i = 1 to n of i³.1936

If I take 1³ + 2³ + 3³ + 4³ + 5³, however high I want to go, that is going to be n × n + 1/ 2².1942

Let us see b as a constant, if c is a constant and if I have the summation as i goes from 1 to n,1963

I have just plain old c, no i anywhere, that is just equal to n × c.1977

It is just telling me, i = 1, let us say n is 10, that means add c 10 times.1985

C + c + c + c and another 5×, that is 10 × c, n × c.1991

This is just a formula for, when I see this, I can substitute this.1999

That is all we are doing here.2005

It is just writing down some formulas that we can use, when we actually start evaluating these integrals using the definition.2007

Another thing that we can do is 1 to n c × ai, if I have some expression that involves a c, I can pull the constant out in front of it.2015

The same that I do, when I distribute.2029

When I factor things out, I can factor out this c.2032

Because this is just ca sub 1, ca sub 2 + ca sub 3.2035

They all involve c, just pull the c out.2041

It equal c × the sum i = 1 to n of a sub i. 2045

What this means is that you can pull a constant out from under the summation symbol.2052

Let us do the sum as i go from 1 to n.2089

If we have a sub i + b sub i, let us say + or -, I can separate these out.2095

I can write this as, the sum i go from 1 to n of a sub i, + or -, depending on whether it is + or -.2102

In other words, the summation distributes over both.2108

You can think of it that way, b sub i 1 to n.2113

Properties of the definite integral.2120

Let us go to properties of the definite integral.2123

These are all going to be very familiar because integration is just a fancy form of summation.2136

The summation symbols, everything that we do for summation, we can do for the integral.2142

The integral from b to a of f(x) dx is equal to the negative of the integral from a to b of f(x) dx.2154

In other words, if I switch the order of integration, if I do here, my Δx is b – a/ n.2166

Here it is b – a/ n, here it is going to be a – b/ n.2177

If I integrate from b to a, instead of from a to b, all I do is switch the sign of the integral.2184

That is it, very simple, that is all.2188

In calculus, we are adding from left to right.2193

We have some function and we have broken up into a bunch of rectangles.2199

If I add in this direction from a to b, I get a number.2202

If I decide to add in this direction, I get the negative of that number.2206

That is all this is saying, switch the upper and lower limit, change the sign.2210

The integral from a to a of f(x) dx that is equal to 0.2219

The integral from a to b of cdx, c is any constant, that is just equal to c × b – a.2227

All this says is, if I have a constant function from a to b, the integral, it is just c, 2239

the value of c which is this height × this distance, the area.2251

Or it could be negative because it is the integral, integral.2257

Now we are thinking, areas can now be negative.2260

If we introduce this notion of a negative area, it is not a problem.2263

Some further properties, the integral from a to b of c × f(x) dx, you can pull the constant out, c × the integral from a to b of f(x) dx.2269

The integral from a to b of f(x) + g(x) dx, the integral of a sum is equal to the sum of the individual integrals.2288

In other words, the integral sign distributes over both.2301

I will just write up the whole thing, of f(x) dx + the integral from a to b of g(x) dx.2311

The integral from a to c of f(x) dx = the integral from a to b of f(x) dx + the integral from b to c of f(x) dx.2326

This last one says I can actually break this up.2341

If I have a here and if I have c here, if i have b here, and if that is my function, 2345

the integral from a to c is just the integral from a to b + the integral from b to c.2353

That is all I’m doing, just adding them up.2358

If f(x) is greater than or equal to 0 on the interval ab, then the integral of f(x) dx is greater than or equal to 0.2370

Very simple, nothing happening here.2391

Integration is something called an operator.2396

An operator is just a fancy word for do something to this function, in other words, operate on it.2403

If I give you a function f(x) and I say integrate it, integrate it means do something to it.2408

Take this function through a series of steps and spit out a number at the end.2412

When you see f(x) is greater than or equal to 0, you already know from years and years, elementary, junior high school math, 2418

that whenever you have an quality or an inequality, as long as you do the same thing to both sides of the quality or the inequality,2426

you retain that equality or inequality.2436

Here, f(x) is greater than or equal to 0.2439

If I integrate the left side, integrate the right side, I get the left side of the integral of f(x) dx.2441

The integral that of 0 is just a bunch of 0 added together.2449

It is 0, it stays.2453

Think about it that way.2456

When you see an equality or inequality, you can integrate both sides.2458

It retains the equality or inequality, it retains the relationship.2463

Just like if you take the logarithm of both sides, if you exponentiate both sides, if you multiply both sides by 2,2467

if you divide both sides by 5, as long as you do it to both sides, you are fine.2472

Treat the integral as some operator, as some thing that you do.2477

If f(x) is greater than or equal to g(x) on ab, then exactly what you think.2484

The interval from a to b of f(x) is greater than or equal to the integral from a to b of g(x).2499

This is greater than that so the integral is greater than integral, very simple.2508

If m is less than or equal to f(x), less than or equal to M on ab,2516

then m × b - a less than or equal to the integral a to b, f(x) dx less than or equal to m × b – a.2532

If some function on this interval happens to have a lower bound and an upper bound, 2548

then the integral of that function is going to be greater than the lower bound × the length of the interval.2555

It is going to be less than the upper bound × the length of the interval.2563

All we have done here is take the integral of this, the integral of that, the integral of that.2567

The integral of that, that is the symbol.2571

The integral of a constant m dx, we said it is equal to m × b – a.2574

The integral of M is M × b – a.2579

Just integrate, integrate, solve for the things you can solve and leave those that you cannot, as the symbol.2587

Thank you so much for joining us here at www.educator.com.2596

We will see you next time, bye.2599