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WOMAN: In this interactive,

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we explore the trapezoidal estimate.

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Here we're looking at the graph

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of y = x squared + 1

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and attempting to approximate

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the area underneath this graph

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between x = 0 and x = 5

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and we're doing this

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using the trapezoidal estimate,

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which involves creating

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a number of trapezia

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to approximate the area.

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Currently n is set to 5,

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which means we've
divided the interval

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into five equal width trapezia.

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Each trapezium has a width of 1,
as we see.

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As we increase n,

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we're obviously increasing
the number of trapezia,

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hence decreasing the width
of each trapezia

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and also seeing that the area

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estimated by the trapezia

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is in fact getting closer and closer

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to the area of the curve.

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This is becoming quite difficult
to see quite quickly on the graph

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given that the trapezia

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creates such a good approximation,

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but we can certainly see
from the numerical calculations

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in the bottom left

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that the total area of the trapezia

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is indeed getting closer and closer

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to the actual area
underneath the graph.

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This approximation,
that is the trapezoidal estimate,

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is a much better approximation

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than either the right endpoint
or the left endpoint

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or in fact the midpoint estimate.

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In fact, the trapezoidal estimate

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is the average of the left endpoint
and the right endpoint estimates

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and you'll have noted

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if you've already had a look
at those interactives

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that the left endpoint estimate

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gives an approximation that's
slightly underneath the actual area

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and the right endpoint estimate

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gives an approximation that's
slightly above the actual area

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and so an average of those two areas
is much more accurate

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and the average of those two areas

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is in fact equal to this
trapezoidal estimate.