WOMAN: In this interactive,
we explore the trapezoidal estimate.
Here we're looking at the graph
of y = x squared + 1
and attempting to approximate
the area underneath this graph
between x = 0 and x = 5
and we're doing this
using the trapezoidal estimate,
which involves creating
a number of trapezia
to approximate the area.
Currently n is set to 5,
which means we've divided the interval
into five equal width trapezia.
Each trapezium has a width of 1, as we see.
As we increase n,
we're obviously increasing the number of trapezia,
hence decreasing the width of each trapezia
and also seeing that the area
estimated by the trapezia
is in fact getting closer and closer
to the area of the curve.
This is becoming quite difficult to see quite quickly on the graph
given that the trapezia
creates such a good approximation,
but we can certainly see from the numerical calculations
in the bottom left
that the total area of the trapezia
is indeed getting closer and closer
to the actual area underneath the graph.
This approximation, that is the trapezoidal estimate,
is a much better approximation
than either the right endpoint or the left endpoint
or in fact the midpoint estimate.
In fact, the trapezoidal estimate
is the average of the left endpoint and the right endpoint estimates
and you'll have noted
if you've already had a look at those interactives
that the left endpoint estimate
gives an approximation that's slightly underneath the actual area
and the right endpoint estimate
gives an approximation that's slightly above the actual area
and so an average of those two areas is much more accurate
and the average of those two areas
is in fact equal to this trapezoidal estimate.