Definite integrals (part II) More on why the antiderivative and the area under a curve are essentially the same thing.
Definite integrals (part II)
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- Welcome back.
- So where I left off, we said that we had this, I guess you
- could call it, equation or this function, although I didn't
- write it with the function notation, where I said, the
- distance is equal to 16 t squared, and I graphed it, it's
- like a parabola, right, for positive time.
- And then we said, well, the velocity, if we know the
- distance, the velocity is just the change of the distance
- with respect to time.
- It's just, the velocity is always changing, you can't
- just take the slope, you actually have to take
- the derivative, right?
- So we took the derivative with respect to time of this
- function, or this equation, and we got 32t, and this
- is the velocity.
- And then we graphed it.
- And then I asked a question.
- I was like, well, we want to figure out, if we were given
- this, if we were given just this, and I asked you, what is
- the distance that this object travels after time, you
- know, after 10 seconds?
- Let's, you know, let's say this t0 is equal to 10 seconds.
- I want to know how far is this thing gone after 10 seconds.
- And let's say you didn't know that you could just take the
- antiderivative, let's say we didn't know this at all, and
- let's say you didn't know that you could just take the
- antiderivative, because we just showed that, you know, the
- derivative of distance is velocity, so the antiderivative
- of velocity is distance.
- So let's say you couldn't just take the antiderivative.
- What's a way that you could start to try to approximate
- how far you've traveled after, say, 10 seconds?
- Well [? as I said, ?]
- you graph this, and you say, let's assume over some
- change in time, velocity is roughly constant, right?
- Let's say velocity is right here.
- So you could approximate how far you travel over that small
- change in time by multiplying that change in time, let's say
- that's like, you know, a millionth of a second, times
- the velocity at roughly that time, or maybe even the average
- velocity over that time, and you'd get the distance you've
- traveled over that very small fraction of time, right?
- But if you look at it visually, that also happens to be the
- area of this rectangle, right?
- And what we said, is if you want to know how far you travel
- after 10 seconds, you just draw a bunch of these rectangles,
- and you sum up the area, right?
- And you could imagine, and you don't have to imagine, it's
- actually true, the smaller the bases of these rectangles, and
- the more of these rectangles you have, the more accurate
- your approximation will be, and you'll approach 2 things.
- You'll approach the area under this curve, right, almost the
- exact area under this curve, and you'd also get almost the
- exact value of the distance after, say, 10 seconds
- in this case, right?
- But 10 didn't have to be an exact number.
- It could have been a variable.
- So this is something pretty interesting.
- All of a sudden, we see that the antiderivative
- is pretty darn similar to the area under the curve.
- And it actually turns out that they're the same thing.
- And this is where I'm going to teach you the
- indefinite integral.
- So the indefinite integral, I don't know how comfortable you
- are with summation, I remember the first time l learned
- calculus, I wasn't that comfortable with summation, but
- it's really, all the indefinite integral, is is you can kind of
- view it as a sum, right?
- So now, you'll maybe understand a little bit more why this
- symbol looks kind of like a sigma.
- That's actually how I view it.
- And please look it up so you can see properly
- drawn integrals.
- But in this case, the indefinite integral is just
- saying, well, I'm going to take the sum from t equals 0, right,
- so from t equals 0, to let's say in this example, t equals
- 10, right, because I said 10.
- From t equals 0 to t equals 10.
- and I'm going to take the sum of each of the heights, the
- height at any given point, which is the velocity.
- And then, what's the formula for the velocity?
- It's 32t and then I'm at times the base at each
- of these rectangles, dt.
- And so this is the definite integral.
- The definite integral is literally, and they never do
- this in math texts, and that's what always kind of confused
- me, is that you can kind of view it like a sum, like this.
- It's kind of the sum of each of these rectangles, but it's the
- limit, as-- if these were discrete rectangles, you could
- just do a sum, and you could make the rectangle bases
- smaller and smaller, and have more and more rectangles,
- and just do a regular sum.
- And actually, that's how, if you ever write a computer
- program to approximate an integral, or approximate the
- area under a curve, that's the way a computer program
- would actually do it.
- But the actual indefinite integral says, well, this is a
- sum, but it's the limit as the bases of these rectangles get
- smaller and smaller and smaller and smaller, and we have more
- and more and more of these rectangles.
- So as these dt's approach 0, the number of rectangles
- actually approach infinity.
- So I'm actually going to, I'll do that more rigorously later,
- but I think it's very important to get this intuitive feel
- of just what an integral is.
- It isn't just this voodoo that happens to be there.
- But anyway, so going back to the problem.
- So the integral from-- this is now a definite integral,
- extending from t equals 0 to t equals 10.
- This tells us 2 things.
- This tells us the area of the curve from t equals zero to t
- equals 10, right, it tells us this whole area, and it also
- tells us how far the object has gone after 10 seconds.
- So it's very interesting.
- The indefinite integral tells us 2 things.
- It tells us area, and it also tells us the antiderivative.
- We're already familiar with it as an antiderivative.
- So let me give you another example.
- Actually, maybe I'll stick with this example, but
- I'll clear it a bit.
- Actually, maybe I should erase.
- Erasing might be a good option with this one,
- since it's fairly messy.
- I think you know all this stuff now.
- I just need space.
- Maybe, OK, so we have that indefinite integral.
- And we could actually figure it out, too.
- I mean, well, after t seconds, [UNINTELLIGIBLE].
- So and the way you evaluate an indefinite integral, and let me
- show you that first, is that you figure out the integral.
- So let me just say, let me continue with the
- problem, actually.
- As you can tell, I don't plan much for these presentations.
- So the way you figure out the indefinite integral, is you
- say, and sometimes they won't write t equals 0 to t equals t.
- They'll just say from 0 to 10 of 32t dt.
- And the way you evaluate this, is you figure out the
- antiderivative, and you really don't have to do the plus c
- here, so the antiderivative, we know, is 16t squared, right?
- It's one half t squared times 32.
- So that's 16t squared.
- And we evaluate this at ten, and we evaluate it at 0, and
- then we subtract the difference.
- So we evaluate this at 10, so 16 times 100, right?
- That's evaluated at 10, and then we subtract
- it, evaluate at 0.
- So 16 times 0 is 0.
- So after 10 seconds, we would have gone 1600 feet.
- And also, the area under this curve is 1600.
- So let's use this to do a couple more examples.
- And actually, I want to show you why we do this subtraction.
- Actually, I'm going to do that right now.
- Let me clear it.
- Oh, that's ugly.
- I'll now do it more general, actually.
- Let me draw this twice, once for the distance, and
- once for its derivative.
- So let's say that the distance, yeah, well, let's just say it
- looks something like this.
- Let's say you start at some distance, and then it
- goes off like that.
- So let's say we call this distance b.
- Well, let's just call this, you know, I don't know, 5.
- We start at 5 feet, and then we moved forward from there.
- And this axis is of course time, this axis, maybe I
- shouldn't do 5, because it looks so much like s.
- That's 5, 5 feet.
- And this is the s, or distance, axis.
- And actually, I just looked at the clock.
- I'm running out of time.
- So let me continue this in the next presentation.
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