- [Voiceover] The graph of F is shown below.
A total of 24 right hand rectangles
are shown.
So, what do I mean by right hand rectangles?
So, there's clearly 24 rectangles.
You can count them.
And right hand rectangle means that
for each of these rectangles the height
of the rectangle is defined by the value
of the function on the right hand side
of the rectangle.
So you can see this is the right hand side
of this first rectangle and if you take
the value of the function of that point
that is the height of the rectangle.
A left hand rectangle would define
the height of the rectangle by the value
of the function on the left hand side
of the rectangle.
So, a left handed rectangle's height,
the first rectangle's height would look
like that.
That's what they mean by right
handed rectangle
Fair enough, eight in blue.
We see that. 16 in red. All right.
All 24 of the rectangles have the same width.
Which of the statements below
is or are true?
They give us three expressions
in sigma notation and they say,
like this first one is the sum of the
areas of the blue rectangles.
This is the sum of the areas of
the red rectangles.
This is the sum of the areas of all
of the rectangles.
So, I encourage you now to pause
the video and try to determine on your own,
which of the statements is or are true.
So, I assume you've had a go at it.
Let's just go through each of these
and see whether they make sense.
So this first one, the sum
of the areas of the blue rectangles.
Well, we know we have one, two, three,
four, five, six, seven, eight blue rectangles,
and we're summing from one to eight.
So it seems like we're summing eight
things right over here.
This is one, two, three, four, five,
six, seven, eight.
So, this is looking good right over here.
And then we take F of something
times one half.
So, we're not even looking at this yet.
It looks like this would be the height
of each of the rectangles.
Remember, we're taking the value
of the function on the right hand side
for the height, and this would be
the width.
So does it make sense that the width
of each of these rectangles is one half?
Well, the total distance between X
equals negative five and X equals seven,
that distance is 12.
Five plus seven, that's 12,
and we're dividing it into 24 rectangles
of equal width.
So, if you divide 12 divided by 24
each of these are going to have a width
of one half.
So this is checking out that the one half.
Now let's think about this part.
Let's think about the F of negative five
plus I over two.
So, let's see.
When I is equal to one, so we're going to
take one half times F of negative five
plus one over two.
Right?
I is one. So negative five plus one half
is going to get us to this point right
over here.
F of that is going to be this
distance, this height right over here.
This is consistent with these being
right handed rectangles.
So, this is definitely the case.
When I is equal to one we're
definitely finding this area right over here.
When I is equal to two it's going to be
negative five plus two over two.
So two over two, we're going to add
one, we're going to go over here.
So once again we're doing one half,
which is this right over here.
That's the width times F of negative five
plus two over two, which is F of negative
four, which is this height right over here.
So, once again that is that area.
You can keep following it.
Everytime we're taking the function
this first one is negative five
plus one half, and then for each
increment we're adding a half to
I guess the right hand side is one way
to think about it.
So, this actually makes complete sense.
We're doing this for the first eight
so this is indeed true.
This is the sum of the areas
of the blue rectangles.
Now let's look at this one over here.
The sum of the areas of the red rectangles.
At first this looks pretty interesting.
We're taking the sum of 16 things,
and we do indeed have 16 things
right over here.
We have the width of each of those
16 things or for each of those things
we want to figure out an area,
and it is indeed the case that each
of these has a width of one half.
But what happens when we take F of
negative one plus I over two.
So, we're starting right over here at
negative one.
Negative one plus I over two.
When I equals one we're going
to be at this point right over here,
and F of that is going to be --
You might say, "Hey, isn't that going
to be the height of that rectangle?
When I equals two isn't it going to be
the height of that rectangle?
And when I is equal to three isn't it
going to be the height of that rectangle?"
And that's where we have to be very careful.
They're going to have the same
absolute value, but these are going to be
these are all going to be negative values.
These are all going to be negative
because we see between this value of
our function.
So that looks like between negative
one half all the way to seven our function
is actually negative.
So one way to think about it, you'd be
having negative height so when you
multiply these two things you're going
to get a negative number.
So this whole thing is going to be
a negative number and so you're
essentially going to get the negative of the sums
of the areas of the red rectangles,
but that's not the same thing as the
sum of the areas of the red rectangles.
An area is, at least in the traditional sense,
you would expect to have, if you were
just seeing what's the area how much carpet
you would need to cover this someone would say
that will be a positive value.
But this is going to be a negative version
of that.
So that is not the sum of the areas
of the areas of the red rectangles.
It's the negative of the areas of the red
rectangles. So we'd rule that one out.
Then this last choice.
So, this expression is the sum
of the areas of all the rectangles.
And so this one is going from I equals
one to 24.
So it's literally 24 things.
It's starting here and it just keeps going.
And if this said from I equals one to I equals
eight it would be the first choice,
but then this falls into the problem
again of once we get past, when we get to I
equal nine this thing right over here
will turn negative, and it's going
to give the negative area.
So it's essentially going to net out
this positive area against this negative
area right over here.
So it's not the sum of the areas
of all the rectangles.
It's going to be this area
essentially minus this area right over there.