Introduction to definite integrals Using the definite integral to solve for the area under a curve. Intuition on why the antiderivative is the same thing as the area under a curve.
Introduction to definite integrals
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- Welcome back.
- In this presentation, I actually want to show you how
- we can use the antiderivative to figure out the
- area under a curve.
- Actually I'm going to focus more a little bit more
- on the intuition.
- So let actually use an example from physics.
- I'll use distance and velocity.
- And actually this could be a good review for derivatives,
- or actually an application of derivatives.
- So let's say that I described the position
- of something moving.
- Let's say it's s.
- Let's say that s is equal to, I don't know, 16t squared.
- So s is distance.
- Let me write this in the corner.
- I don't know why the convention is to use s as
- the variable for distance.
- One would think, well actually, I know, why won't they use d?
- Because d is the letter used for differential, I guess.
- So s is equal to distance, and then t is equal to time.
- So this is just a formula that tells us the position, kind of
- how far has something gone, after x many, let's
- say, seconds, right?
- So after like, 4 seconds, we would have gone, let's say
- the distance is in feet, this is in seconds.
- After 4 seconds, we would have gone 256 feet.
- That's all that says.
- And let me graph that as well.
- Graph it.
- That's a horrible line.
- Have to use the line tool, might have better luck.
- It's slightly better.
- Actually, let me undo that too, because I just want to do
- it for positive t, right?
- Because you can't really go back in time.
- For the purposes of this lecture, you can't
- go back in time.
- So that'll have to do.
- So this curve will essentially just be a parabola, right?
- It'll look something like this.
- So actually, if you look at it, I mean you
- could just eyeball it.
- The object, every second you go, it's going a little
- bit further, right?
- So it's actually accelerating.
- And so what if we wanted to figure out what the velocity
- of this object, right?
- This is, let's see, this is d, this is t, right?
- And this is, I don't know if it's clear, but this is
- kind of 1/2 a parabola.
- So this is the distance function.
- What would the velocity be?
- Well the velocity is just, what's velocity?
- It's distance divided by time, right?
- And since this velocity is always changing, we
- want to figure out the instantaneous velocity.
- And that's actually one of the initial uses of what made
- derivatives so useful.
- So we want to find the change, the instantaneous change
- with respect to time of this formula, right?
- Because this is the distance formula.
- So if we know the instant rate of change of distance with
- respect to time, we'll know the velocity, right?
- So ds, dt, is equal to?
- So what's the derivative here?
- It's 32t, right?
- And this is the velocity.
- Maybe I should switch back to, let me write that,
- v equals velocity.
- I don't know why I switched colors, but I'll stick
- with the yellow.
- So let's graph this function.
- This will actually be a fairly straightforward graph to draw.
- It's pretty straight.
- And then we draw the x-axis.
- I'm doing pretty good.
- So this, I'll draw it in red, this is this going
- to be a line, right?
- 32t it's a line with slope 32.
- So it's actually a pretty steep line.
- I won't draw it that steep because I'm going to use
- this for an illustration.
- So this is the velocity.
- This is velocity.
- This is that graph, and this is distance, right?
- So in case you hadn't learned already, and maybe I'll do a
- whole presentation on kind of using calculus for physics, and
- using derivatives for physics.
- But if you have to distance formula, it's derivative
- is just velocity.
- And I guess if you view it the other way, if you
- have the velocity, it's antiderivative is distance.
- Although you won't know where, at what position,
- the object started.
- In this case, the object started at position of 0,
- but it could be, you know, at any constant, right?
- You could have started here and then curved up.
- But anyway, let's just assume we started at 0.
- So the derivative of distance is velocity, the antiderivative
- of velocity is distance.
- Keep that in mind.
- Well let's look at this.
- Let's assume that we were only given this graph.
- And we said, you know, this is the graph of the
- velocity of some object.
- And we want to figure out what the distance is after, you
- know, t seconds, right?
- So this is the t-axis, this is the velocity axis, right?
- So let's say we were only given this, and let's say we didn't
- know that the antiderivative of the velocity function is
- the distance function.
- How would we figure out, how would we figure out what
- the distance would be at any given time?
- Well let's think about it.
- If we have a constant, this red is kind of bloody.
- Let me switch to something more pleasant.
- If we have, over any small period of time, right, or if we
- have a constant velocity, when you have a constant velocity,
- distance is just velocity times time, right?
- So let's say we had a very small time
- fragment here, right?
- I'll draw it big, but let's say this time fragment
- it is really small.
- And let's called this very small time fragment, let call
- this delta t, or dt actually.
- The way I've used dt is like, it's like a change in time
- that's unbelievably small, right?
- So it's like almost instantaneous, but not quite.
- Or you can actually view it as instantaneous.
- So this is how much time goes by.
- You can kind of view this as a very small change in time.
- So if we have a very small change of time, and over that
- very small change in time, we have a roughly constant
- velocity, let's say the roughly constant velocity is this.
- Right, this is the velocity, so say we had over this very small
- change in time, we have this roughly constant velocity
- that's on this graph.
- Actually, let me take do it here.
- We have this roughly constant velocity.
- So the distance that the object travels over the small time
- would be the small time times the velocity, right?
- It would be whatever the value of this red line is, times the
- width of this distance, right?
- So what's another way?
- Visually I kind of did it ahead of time, but
- what's happening here?
- If I take this change in time, right, which is kind of the
- base of this rectangle, and I multiply it times the velocity
- which is really just the height of this rectangle, what
- have I figured out?
- Well I figured out the area of this rectangle, right?
- Right, the velocity this moment, times the change in
- time at this moment, is nothing but the area of
- this very skinny rectangle.
- Skinny and tall, right?
- It's almost infinitely skinny, but it's, we're assuming for
- these purposes it has some very notional amount of width.
- So there we figured out the area of this column, right?
- Well, if we wanted to figure out the distance that you
- travel after, let's say, you know, I don't know, let's say
- t, let's say t sub nought, right?
- This is just a particular t.
- After t sub nought seconds, right?
- Well then, all we would have to do is, we would have to just
- figure, we would just do a bunch of dt's, right?
- You'd do another one here, you'd figure out the area of
- this column, you'd figure out the area of this column, the
- area of this column, right?
- Because each of these areas of each of these columns
- represents the distance that the object travels
- over that dt, right?
- So if you wanted to know how far you traveled after t sub
- zero seconds, you'd essentially get, or an approximation would
- be, the sum of all of these areas.
- And as you got more and more, as you made the dt's smaller
- and smaller, skinnier, skinnier, skinnier.
- And you had more and more and more and more of these
- rectangles, then your approximation will get pretty
- close to, well, two things.
- It'll get pretty close to, as you can imagine, the area
- under this curve, or in this case a line.
- But it would also get you pretty much the exact amount
- of distance you've traveled after t sub nought seconds.
- So I think I'm running into the ten minute wall, so I'm just
- going to pause here, and I'm going to continue this in
- the next presentation.
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