Integrals and Areas
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Recall that the major application of an integral is finding the area under the curve.
This could be expressed as:
b
dx
x
f
)
(
a

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Integrals and Areas
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We know that if we take a summation rather than an integral, we are
approximating the area rather than finding the full area. Caution: The integral
will not be the value of the sum as we are taking boxes of width 1 (not a finely
defined area as done by an integral)
defined area as done by an integral).
Consider a graph that is strictly
decreasing
,
positive
, and
continuous
. We can take
ti l
b
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th
idth
f
b
t
b
1
d
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b
th
a partial area by choosing the width of our box to be 1, and our height to be the
value of the function:
This means that a
2
= f(2), a
3
= f(3) …