Approximating Integrals

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Lecture 19

• Approximating Integrals

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Integral Area
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Estimating Using Rectangles
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Size of Rectangle Base Can Vary
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Area of approximating Rectangle f(c) (b-a)
f(c)
c
a
b
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A partition of an Interval a,bis just a
choice of numbers with
for each i 0.. n-1
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Examples
1,2,3,4,5,9,12 is a partition of
1,12 -5,0,20 is a partition of -5,20
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A marking of the partition is just a choice
of numbers
With for each i
1.. n
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Examples
3,7 is a making of the partition 1, 5,12
since 3 is in 1,5 and 7 is in
5,12 1,2,3,4,5 is a marking of the
partition1,2,3,4,5,6 since 1 is in 1,2, 2
is in 2,3,
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Integral Estimates from Partitions
If P
Is a partition of a,b and
is a marking of P then for any function f defined
on a,b we have an for
given by
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Theorem
As the maximum distance between elements of the
partition goes to zero the estimates converge to
the actual integral
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Regular Partitions
A partition is regular all of the rectangles it
defines have the same Base length. More precisely
is regular if is the
same for each i.
If then nh
b-a or
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Regular Estimates for
Uses the regular partition of a,b into
nsubintervals and the left end point of each
subinterval for the marking
Uses the regular partition of a,b into n
subintervals and the left end point of each
subinterval for the marking
(The Trapezoid Rule) is the average of
and
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Example
Calculate the left regular estimate for
with 5 subdivisions
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90
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Example
Calculate the right regular estimate for
with 5 subdivisions
2
170

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Example
Calculate the trapezoid rule estimate for
with 5 subdivisions
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Average of a Function
If f(x) is a function defined on an interval
a,b then the average value of f on a,b is
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The temperatures every two hours from
midnight(time t 0 hrs) to noon is given in the
table. Estimatethe average temperature over that
time interval using theleft regular, right
regular, and trapezoid estimates.
a 0, b 12, n 6
h 2

The left estimate is
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Similarly the right estimate is
482 532 572602622632 686
The trapezoid rule estimate for the integral is
666 so the trapezoid rule estimates the
averagetemperature as
55.5
Remark The average of the temperatures above is
So the average of the temperatures is not the
same as the average temperature
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