In this interactive,
we explore the left endpoint estimate.
We're going to look at the left endpoint estimate
under the graph of y = x squared + 1
from x = 0 to x = 5.
So at the moment we see
that n is set to 5,
that is, that we're using
five rectangles
to estimate this area.
So given that we're looking
at the area from 0 to 5,
five rectangles,
each have a width of 1
and we can calculate the area
of each of those five rectangles.
You'll see in this interactive
over here where we have 'Sequence',
that's giving us the x coordinates
of each of the boundaries between the rectangles.
The 'Areas of Rectangles'
is, obviously, calculating the area of each rectangle,
so that is, that x value from that sequence
at the left endpoint -
so in this case using the x values of 0, 1, 2, 3 and 4 -
and multiplying them by the heights of the rectangles
at those corresponding positions.
We'll see over here on the left
we're being told the n value and also delta x.
Delta x refers to the width of each rectangle,
which in this case is 1.
We're also being shown the total area of the rectangles
along with what the actual area
under the curve of y = x squared + 1 between 0 and 5 is
to allow us to make a comparison.
So we see that as we increase
the value of n, we're getting
more rectangles, obviously.
Those rectangles
are each getting thinner.
So each rectangle now has a width,
as we can see down here,
of 5 divided by 7 -
we've divided a distance of 5
into 7 equal units.
And we're also seeing each of those x values
that creates the boundaries between each rectangle.
So the first one is, obviously, at five-sevenths,
the next one is at 2 times five-sevenths,
the next one is at 3 times five-sevenths, etc.
So we're getting more rectangles.
We also note that as we increase
the number of rectangles,
the total area of the rectangles
is getting closer to the actual area.
We can see that quite physically
in the diagram above
and we can also see that
in the calculations
that are occurring down here below.
So as we watch
the number of rectangles increase,
we physically see that area
get close to the actual area
and we also see numerically
the area of the left endpoint estimate
get closer and closer and closer to the actual area.