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[music] I want to be able to compute areas
of curved regions.

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For instance, here I've got the graph of y
equals x squared.

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And between 0 and 4, I could ask for some
calculation.

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Some approximation, at least, of the area
above the x axis and below the graph of y

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equals x squared, right.
I want to approximate this area in here.

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Practically, the only thing that I
actually know the area of is rectangles.

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So, I'm going to replace this curved
region with an approximation by

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rectangles.
The first step is to take that interval, 0

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to 4, and cut it up into some smaller
pieces.

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I'll partition the interval from 0 to 4
into few pieces.

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I'll cut it into 3 pieces.
Let's cut it into a piece from 0 to 2 and

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piece from 2 to 3.
And then, one more piece from 3 to 4.

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So, I've partitioned the interval from 0
to 4 into some pieces.

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0 to 2, 2 to 3, 3 to 4.
Now, I'll pick some points inside each of

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those sub intervals, points where I'm
going to end up sampling the function.

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So, in this first interval, I'll pick the
point 1.

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In this second interval from 2 to 3, I'll
pick the point 2.

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And in the third interval from 3 to 4,
I'll pick the point 3.5.

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Right?
So, I'm picking some sample points in each

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of the pieces of my partition.
Now, I'll build a skyscraper picture.

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A, a bunch of rectangles that approximates
my curved region.

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So, I'll sample my function at these
sample points.

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And I'm going to draw this sort of
skyscraper picture.

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So, I'm going to draw a box over the first
interval of height, whatever the sample

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was.
So, there is the height is given by the

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function's value at the sample point.
In my second interval from 2 to 3, I

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sampled the 2.
And I'll draw a little rectangle there.

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And in my third interval from 3 to 4, I
sampled at 3.5.

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And then, I'll draw one more rectangle
right there.

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So now, my three sub intervals gave rise
to three rectangles.

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Now, what's the area of those rectangles?
Well, this first rectangle has a height of

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1 square and a width of 2.
So, this first wide, but not very tall

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rectangle, has area 2.
This rectangle here has a height of 2

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squared and a width of 1.
So, this rectangle has area 4.

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And this last rectangle has a height of
3.5 squared and a width of 1.

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So, it has area 3.5 squared, which is
12.25 square units.

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And now, if I add 2 plus 4 plus 12.25, I
get then an area of 18.25.

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And that's suppose to be an approximation
to the area of the curved region.

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Right, I replaced this curved region by
these three rectangles.

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And I can calculate the areas of these
three rectangles.

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It's going to be about 18.25 and that
should be not so far off from the true

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area of this curved region.
So, that's an approximation, but we can do

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better.
If we want a better approximation to the

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area under this curve, I'll just cut the
interval between 0 and 4 up into smaller

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pieces.
I could, in this case, consistently choose

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the left end point to sample that and I
can draw a whole bunch of rectangles now.

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And I could approximate the area under the
curve just by calculating the area of

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these rectangles.
And by taking more and more rectangles

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thinner and thinner, I'll get an
increasing good approximation to the true

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area under this curve.
I can also over estimate the curved

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region's area.
I could also try to over estimate this.

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I could cut this interval from 0 to 4 into
a bunch of small intervals.

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And then, choose the right end point of
each of each of these intervals to do my

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sampling at.
And then, build the sort of skyscraper

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picture.
Again, sampling at the right hand end

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point of each of my little tiny sub
intervals and I can build this little

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skyscraper picture.
And by computing the area of the

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rectangles.
Alright, the areas of these rectangles are

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a pretty good over-approximation to the
area under the curve.

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I'm going to give a name to all these
kinds of approximations.

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The name that were given to these things
is a Riemann Sum, right?

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Riemann was a mathematician.
So, what's the procedure for setting up a

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Riemann Sum?
Well it's really a multi-step process,

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right?
Here's a picture that sort of summarizes

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the whole story.
I'm trying to calculate the area under the

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curve between the points a and b and I'm
doing that approximation by approximating

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the curved area with these rectangles.
And I can, of course, calculate the areas

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of these rectangles and add them up.
But what's the first step?

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Well, the first step is to partition this
interval a to b, so that I can decide

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where to build my little skyscrapers.
Alright, so the first step here is to

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partition the interval a to be and what
that involves is choosing x not, which is

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just a.
And x sub n which is b.

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And then, choosing x1, x2, and so forth in
between a and b, alright?.

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In this picture, I've chosen x1, x2, x3,
x4 And I've cut it up into one, two,

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three, four, five pieces.
But, of course, you can choose how many

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pieces to cut it up into.
The point is just to choose how many

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pieces to cut it up into, and then, choose
where you're going to make those, those

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cuts in the partition.
The next step is to choose sample points

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inside each of those sub intervals.
Yeah, after I've decided where to cut up

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my interval a to b, I've chosen the x
sub-I's.

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Then, I introduce these blue sample
points, right, the x sub i stars.

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So, the next step is to choose sample
points x sub i star, and those sample

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points will then be used to determine the
heights of these rectangles.

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I'm going to be evaluating the function at
those sample points.

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Now, I can write down a formula for the
area.

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Well the area is approximately the areas
of all these rectangles added up.

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Now, how big are each of these rectangles?
The heights of these rectangles are coming

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from the function evaluated at these
sample points and the width has to do with

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how far apart the red cut points are.
So, we can write down a, a formula for

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that, right?
I'm going to write down a formula for the

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area.
So, the area is approximately, and this is

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really Riemann Sum, right?
It's the sum, i goes from 1 to n.

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So, I'm going to add up all of the
rectangles.

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And now, I've got to write down the area
of the I for rectangle or the height of

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the I for rectangle, is the function
evaluated at the sample point.

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And the width of the ith for rectangle is
x sub i minus x sub i minus 1.

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Right?
If I take the i cut point and subtract the

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previous cut point, the difference there
will tell me the width of that rectangle.

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Let me show you on the picture again here.
Alright.

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So, the height of these rectangles are
coming from evaluating the function at the

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sample points and the width of, say, the
1th rectangle, the first rectangle, is x

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sub 1 minus x sub 0.
That's the width times height, that gives

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me the area.
And if I add up the areas of all of the

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green rectangles, then I get an
approximation for the total area under the

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curve.
And just to make it easy to talk about

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these things, there's some specific names
that I want to give to particular kinds of

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Riemann Sums.
For instance, if we pick our ith sample

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point, so x sub i star, to be the left
hand endpoint, x sub i minus 1.

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Then, I can write down my Riemann Sum like
this.

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It's the sum over all the rectangles of f
evaluated at the left hand end point,

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that's sample point, times the width of
that particular rectangle.

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This is called a left Riemann Sum, just
because I've chosen my sample points to be

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in the left hand side of the interval.
Similarly, right?

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There's a right Riemann Sum where I choose
my sample point to be on the right hand

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side, right?
So, this would normally be x sub i star,

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but my sample point is my right hand
endpoint.

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So, it's just x sub i.
I'm going to call this thing a right

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Riemann Sum.