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In this interactive,

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we explore the left
endpoint estimate.

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We're going to look
at the left endpoint estimate

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under the graph of y = x squared + 1

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from x = 0 to x = 5.

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So at the moment we see

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that n is set to 5,

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that is, that we're using

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five rectangles

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to estimate this area.

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So given that we're looking

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at the area from 0 to 5,

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five rectangles,

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each have a width of 1

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and we can calculate the area

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of each of those five rectangles.

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You'll see in this interactive

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over here where we have 'Sequence',

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that's giving us the x coordinates

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of each of the boundaries
between the rectangles.

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The 'Areas of Rectangles'

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is, obviously, calculating
the area of each rectangle,

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so that is, that x value
from that sequence

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at the left endpoint -

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so in this case using the x values
of 0, 1, 2, 3 and 4 -

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and multiplying them
by the heights of the rectangles

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at those corresponding positions.

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We'll see over here on the left

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we're being told the n value
and also delta x.

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Delta x refers to the width
of each rectangle,

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which in this case is 1.

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We're also being shown
the total area of the rectangles

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along with what the actual area

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under the curve of y = x squared + 1
between 0 and 5 is

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to allow us to make a comparison.

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So we see that as we increase

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the value of n, we're getting

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more rectangles, obviously.

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Those rectangles

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are each getting thinner.

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So each rectangle now has a width,

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as we can see down here,

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of 5 divided by 7 -

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we've divided a distance of 5

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into 7 equal units.

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And we're also seeing
each of those x values

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that creates the boundaries
between each rectangle.

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So the first one is,
obviously, at five-sevenths,

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the next one
is at 2 times five-sevenths,

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the next one is at
3 times five-sevenths, etc.

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So we're getting more rectangles.

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We also note that as we increase

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the number of rectangles,

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the total area of the rectangles

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is getting closer to the actual area.

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We can see that quite physically

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in the diagram above

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and we can also see that

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in the calculations

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that are occurring down here below.

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So as we watch

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the number of rectangles increase,

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we physically see that area

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get close to the actual area

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and we also see numerically

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the area of the left
endpoint estimate

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get closer and closer and closer
to the actual area.