[music] Welcome back to Calculus One, and welcome to week 11 of our time together.
Well, this week marks a dramatic change from what's been going on in the course
for the last two months. Pretty much every single week has involved
derivatives in some form, and this week is a sudden exception.
We're really not going to be looking at derivatives this week at all.
Instead, this week, we're going to be looking at integration and area.
Though, you might think that what we're going to be doing is computing a bunch of
areas, and, and that's true, but it's be more precise to say that this week is
about defining the very notion of area. Think back way at the beginning of course
when we first introduced the derivative. You might have said the derivative was
computing instantaneous velocity. But that's not the whole story, right?
The derivative doesn't just let us compute instantaneous velocity.
The derivative defines what the very idea of instantaneous velocity means.
In the same way, this week, we're going to cover the integral.
And the integral does compute area, but it computes area in the same way that the
derivative computes instantaneous velocity, right?
The integral will define what area even means.
We're going to try and write down an actual definition of area under the graph
of a function. Now, how does this fit in with the rest of
the course? Well, last week, in Week 10, we looked at
anti-differentiation. And this week, in Week 11, we're looking
at integrals and area. And next week, in Week 12, we'll introduce
the fundamental theorem of Calculus. And it's that fundamental theorem of
Calculus in Week 12, which will tie together the material in Week 10,
anti-differentiation, and the material this week, integration, and reveal that
anti-differentiation and integration are somehow related.
So unfortunately, to really understand, you know, why for instance we did this
stuff in Week 10, you've got to wait until Week 12.
And this week, we're sort of stuck in the middle where we're just defining the basic
notions of what area means. We're also going to be introducing sigma
notation and doing some summation. But that's really necessary just for being
able to write down the definition of integral.
Now, I should also say that you've got a midterm that you're probably still working
on, and that's great. And if you've got questions about
anything, that midterm, the material from this week on integration, where we're
going, where we've been, anything about the course at all, please contact us,
right? Send us a message, or write to us in the
forum. We want to make sure that all of us can
make it past that finish line together.