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Current time:0:00Total duration:8:31

AP Calc: LIM‑5 (EU), LIM‑5.B (LO), LIM‑5.B.2 (EK)

In the last video, we attempted
to approximate the area under a curve by constructing
four rectangles of equal width and using the left boundary of
each rectangle, the function evaluated at the left boundary,
to determine the height, and we came up with
an approximation. What I want to do in this video
is generalize things a bit using the exact same
method, but doing it for an arbitrary function
with arbitrary boundaries and an arbitrary
number of rectangles. So let's do it. So I'm going to draw the
diagram as large as I can to make things
as clear as possible. So that's my y-axis. And this right over
here is my x-axis. Let me draw an
arbitrary function. So let's say my function
looks something like that. So that is y is equal to f of x. And let me define my boundaries. So let's say this right
over here is x equals a. And this right over
here is x equals b. So this is b. And I'm going to
use n rectangles, and I'm going to use
the function evaluated at the left boundary
of the rectangle to determine its height. So, for example, this
will be rectangle one. I'm going to evaluate
what f of a is. So this right over
here is f of a. And then I'm going to
use that as the height of my first rectangle. So just like that. So rectangle number
one looks like this. And I'll even number it. Rectangle one looks
just like that. And just to have
a convention here, because I'm going to want to
label each of the x values at the left boundary, so
we'll say a is equal to x0. a is equal to x0. So we could also call this
point right over here x0, that x value. And then we go to
the next rectangle. And we could call this one
right over here, this x value, we'll call it x1. It's the left boundary
of the next rectangle. If we evaluate f of x1, we get
this value right over here. This right over here is f of
x1, so it tells us our height. And we want an equal
width to the previous one. We'll think about what the width
is going to be in a second. So this right over here
is our second rectangle that we're going to use
to approximate the area under the curve. That's rectangle number two. Let's do rectangle number three. Well, rectangle number
three, the left boundary, we're just going to
call that x sub 2. And its height is going
to be f of x sub 2. And its width is going to be the
same width as the other ones. I'm just eyeballing
it right over here. So this is rectangle
number three. And we're going to continue
this process all the way until we get to
rectangle number n. So this is the n-th
rectangle right over here, the n-th rectangle. And what am I going to label
this point right over here? Well, we already see a pattern. The left boundary of the
first rectangle is x sub 0. The left boundary of the
second rectangle is x sub 1. The left boundary of the
third rectangle is x sub 2. So the left boundary
of the n-th rectangle is going to be x sub n minus 1. Whatever the
rectangle number is, the left boundary is x
sub that number minus 1. And this is just based on the
convention that we've defined. Now, the next thing that
we need to do in order to actually calculate
this area is think about what is the width? So let's call the width of
any of these rectangles-- and for these purposes, or
the purpose of this example, I'm going to assume
that it's constant, although you can do these
sums where you actually vary the width of the rectangle. But then it gets a
little bit fancier. So I want it equal width. So I want delta x
to be equal width. And to think about
what that has to be, we just have to think, what's
the total width that we're covering? Well, the total distance here
is going to be b minus a, and we're just going to divide
by the number of rectangles that we want, the number
of sections that we want. So we want to divide by n. So if we assume this
is true, and then we assume that a is
equal to x0, and then x1 is equal to x0 plus delta x,
x2 is equal to x1 plus delta x, and we go all the way to xn is
equal to xn minus 1 plus delta x, then we've essentially set
up this diagram right over here. b is actually going
to be equal xn. So this is xn. It's equal to xn
minus 1 plus delta x. So now I think we've set up
all of the notation and all the conventions in order
to actually calculate the area, or our
approximation of the area. So our approximation,
approximate area, is going to be equal to what? Well, it's going to be the
area of the first rectangle-- so let me write this down. So it's going to be rectangle
one-- so the area of rectangle one-- so rectangle one plus
the area of rectangle two plus the area of rectangle
three-- I think you get the point here-- plus all the
way to the area of rectangle n. And so what are
these going to be? Rectangle one is going to be
its height, which is f of x0 or f of a. Either way. x0 and a
are the same thing. So it's f of a
times our delta x, times our width, our
height times our width. So times delta-- actually
I can write as f of x0, I wanted to write-- f
of x0 times delta x. What is our height
of rectangle two? It's f of x1 times delta x. What's our area of
rectangle three? It's f of x2 times delta x. And then we go all
the way to our area. We're taking all the sums,
all the way to rectangle n. What's its area? It's f of x sub n minus 1. Actually, that's a
different shade of orange. I'll use that same shade. It is f of x sub n
minus 1 times delta x. And we're done. We've written it in
a very general way. But to really make
us comfortable with the various
forms of notation, especially the types
of notation you might see when people are
talking about approximating the areas or sums
in general, I'm going to use the
traditional sigma notation. So another way we could
write this, as the sum, this is equal to the
sum from-- and remember, this is just based on the
conventions that I set up. I'll let i count which rectangle
we're in, from i equals 1 to n. And then we're going to
look at each rectangle. So the first rectangle,
that's rectangle one. So it's going to be f of-- well,
if we're in the i-th rectangle, then the left boundary
is going to be x sub i minus 1 times delta x. And so here, right over
here, is a general way of thinking about approximating
the area under a curve using rectangles, where the
height of the rectangles are defined by
the left boundary. And this tells us it's
the left boundary. And we see for each, if this
is the i-th rectangle right over here, if this is rectangle
i, then this right over here is x sub i minus 1, and
this height right over here is f of x sub i minus 1. So that's all we did right
over there times delta x. And then you sum of these
from the first rectangle all the way to the end. So hopefully that makes you
a little bit more comfortable with this notation. We're not doing
anything different than we did in this first
video, which was hopefully fairly straightforward for you. We have just generalized it
using a little bit more mathy notation.