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Welcome back.

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So where I left off, we said that we had this, I guess you

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could call it, equation or this function, although I didn't

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write it with the function notation, where I said, the

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distance is equal to 16 t squared, and I graphed it, it's

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like a parabola, right, for positive time.

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And then we said, well, the velocity, if we know the

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distance, the velocity is just the change of the distance

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with respect to time.

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It's just, the velocity is always changing, you can't

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just take the slope, you actually have to take

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the derivative, right?

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So we took the derivative with respect to time of this

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function, or this equation, and we got 32t, and this

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is the velocity.

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And then we graphed it.

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And then I asked a question.

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I was like, well, we want to figure out, if we were given

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this, if we were given just this, and I asked you, what is

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the distance that this object travels after time, you

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know, after 10 seconds?

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Let's, you know, let's say this t0 is equal to 10 seconds.

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I want to know how far is this thing gone after 10 seconds.

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And let's say you didn't know that you could just take the

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antiderivative, let's say we didn't know this at all, and

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let's say you didn't know that you could just take the

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antiderivative, because we just showed that, you know, the

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derivative of distance is velocity, so the antiderivative

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of velocity is distance.

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So let's say you couldn't just take the antiderivative.

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What's a way that you could start to try to approximate

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how far you've traveled after, say, 10 seconds?

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Well [? as I said, ?]

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you graph this, and you say, let's assume over some

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change in time, velocity is roughly constant, right?

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Let's say velocity is right here.

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So you could approximate how far you travel over that small

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change in time by multiplying that change in time, let's say

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that's like, you know, a millionth of a second, times

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the velocity at roughly that time, or maybe even the average

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velocity over that time, and you'd get the distance you've

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traveled over that very small fraction of time, right?

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But if you look at it visually, that also happens to be the

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area of this rectangle, right?

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And what we said, is if you want to know how far you travel

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after 10 seconds, you just draw a bunch of these rectangles,

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and you sum up the area, right?

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And you could imagine, and you don't have to imagine, it's

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actually true, the smaller the bases of these rectangles, and

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the more of these rectangles you have, the more accurate

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your approximation will be, and you'll approach 2 things.

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You'll approach the area under this curve, right, almost the

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exact area under this curve, and you'd also get almost the

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exact value of the distance after, say, 10 seconds

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in this case, right?

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But 10 didn't have to be an exact number.

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It could have been a variable.

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So this is something pretty interesting.

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All of a sudden, we see that the antiderivative

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is pretty darn similar to the area under the curve.

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And it actually turns out that they're the same thing.

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And this is where I'm going to teach you the

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indefinite integral.

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So the indefinite integral, I don't know how comfortable you

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are with summation, I remember the first time l learned

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calculus, I wasn't that comfortable with summation, but

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it's really, all the indefinite integral, is is you can kind of

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view it as a sum, right?

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So now, you'll maybe understand a little bit more why this

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symbol looks kind of like a sigma.

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That's actually how I view it.

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And please look it up so you can see properly

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drawn integrals.

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But in this case, the indefinite integral is just

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saying, well, I'm going to take the sum from t equals 0, right,

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so from t equals 0, to let's say in this example, t equals

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10, right, because I said 10.

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From t equals 0 to t equals 10.

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and I'm going to take the sum of each of the heights, the

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height at any given point, which is the velocity.

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And then, what's the formula for the velocity?

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It's 32t and then I'm at times the base at each

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of these rectangles, dt.

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And so this is the definite integral.

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The definite integral is literally, and they never do

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this in math texts, and that's what always kind of confused

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me, is that you can kind of view it like a sum, like this.

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It's kind of the sum of each of these rectangles, but it's the

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limit, as-- if these were discrete rectangles, you could

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just do a sum, and you could make the rectangle bases

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smaller and smaller, and have more and more rectangles,

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and just do a regular sum.

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And actually, that's how, if you ever write a computer

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program to approximate an integral, or approximate the

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area under a curve, that's the way a computer program

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would actually do it.

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But the actual indefinite integral says, well, this is a

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sum, but it's the limit as the bases of these rectangles get

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smaller and smaller and smaller and smaller, and we have more

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and more and more of these rectangles.

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So as these dt's approach 0, the number of rectangles

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actually approach infinity.

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So I'm actually going to, I'll do that more rigorously later,

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but I think it's very important to get this intuitive feel

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of just what an integral is.

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It isn't just this voodoo that happens to be there.

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But anyway, so going back to the problem.

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So the integral from-- this is now a definite integral,

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extending from t equals 0 to t equals 10.

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This tells us 2 things.

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This tells us the area of the curve from t equals zero to t

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equals 10, right, it tells us this whole area, and it also

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tells us how far the object has gone after 10 seconds.

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Right?

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So it's very interesting.

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The indefinite integral tells us 2 things.

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It tells us area, and it also tells us the antiderivative.

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Right?

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We're already familiar with it as an antiderivative.

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So let me give you another example.

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Actually, maybe I'll stick with this example, but

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I'll clear it a bit.

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Actually, maybe I should erase.

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Erasing might be a good option with this one,

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since it's fairly messy.

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I think you know all this stuff now.

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I just need space.

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Maybe, OK, so we have that indefinite integral.

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And we could actually figure it out, too.

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I mean, well, after t seconds, [UNINTELLIGIBLE].

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So and the way you evaluate an indefinite integral, and let me

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show you that first, is that you figure out the integral.

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So let me just say, let me continue with the

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problem, actually.

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As you can tell, I don't plan much for these presentations.

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So the way you figure out the indefinite integral, is you

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say, and sometimes they won't write t equals 0 to t equals t.

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They'll just say from 0 to 10 of 32t dt.

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Right?

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And the way you evaluate this, is you figure out the

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antiderivative, and you really don't have to do the plus c

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here, so the antiderivative, we know, is 16t squared, right?

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It's one half t squared times 32.

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So that's 16t squared.

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And we evaluate this at ten, and we evaluate it at 0, and

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then we subtract the difference.

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So we evaluate this at 10, so 16 times 100, right?

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That's evaluated at 10, and then we subtract

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it, evaluate at 0.

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So 16 times 0 is 0.

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So after 10 seconds, we would have gone 1600 feet.

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And also, the area under this curve is 1600.

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So let's use this to do a couple more examples.

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And actually, I want to show you why we do this subtraction.

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Actually, I'm going to do that right now.

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Let me clear it.

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Oh, that's ugly.

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I'll now do it more general, actually.

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Let me draw this twice, once for the distance, and

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once for its derivative.

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So let's say that the distance, yeah, well, let's just say it

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looks something like this.

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Let's say you start at some distance, and then it

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goes off like that.

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Right?

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So let's say we call this distance b.

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Well, let's just call this, you know, I don't know, 5.

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Right?

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We start at 5 feet, and then we moved forward from there.

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And this axis is of course time, this axis, maybe I

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shouldn't do 5, because it looks so much like s.

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That's 5, 5 feet.

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And this is the s, or distance, axis.

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And actually, I just looked at the clock.

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I'm running out of time.

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So let me continue this in the next presentation.

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