In this interactive,
we explore the midpoint estimate.
Here, we're looking at the graph
of y = x squared + 1
and attempting to find an estimate
for the area under this curve
between x = 0 and x = 5.
At the moment, n is set to 5.
That is, we've divided this region
into five rectangles.
That then makes the width of each rectangle,
which is delta x down here,
equal to 1, that is, 5 divided by five,
and each of the boundaries of the rectangles
can also be obtained using the width of this rectangle.
So the first boundary is at x = 1.
The next boundary is at 2 times the width of the rectangles.
That is, x = 2.
The next boundary is at 3 times the width of the rectangles,
that is, x = 3.
And as we increase the value of n,
we see the number
of rectangles increasing.
That makes their width decrease
so now we have the width
of each of the seven rectangles
is equal to 5 divided by 7
and we also see the boundaries changing,
so the boundary between the first and the second rectangle
is now at five-sevenths -
that is, x1 is at five-sevenths,
x2 is at 2 times five-sevenths,
which is ten-sevenths, etc.
And down here we're seeing
the areas of each of these rectangles being calculated
using the height at the midpoint between each of these boundaries
to calculate the area.
Down here we're seeing the total area of the rectangles shown
and the actual area underneath this graph.
And we should see
that as we increase the value of n,
firstly we see
more and more rectangles
and we can also see that
the area of the rectangles is getting
closer and closer and closer
to the area of the actual area
and we can see that both graphically
and also by looking at the
numeric calculations down below here.