CHAPTER 4 SECTION 4.3 RIEMANN SUMS AND DEFINITE INTEGRALS

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CHAPTER 4SECTION 4.3RIEMANN SUMS AND DEFINITE
INTEGRALS
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Riemann Sum

• Partition the interval a,b into n
  subintervalsa x0 lt x1 lt xn-1lt xn b
• Call this partition P
• The kth subinterval is ?xk xk-1 xk
• Largest ?xk is called the norm, called ?
• If all subintervals are of equal length, the norm
  is called regular.
• Choose an arbitrary value from each subinterval,
  call it

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Riemann Sum

• Form the sumThis is the Riemann sum
  associated with
• the function f
• the given partition P
• the chosen subinterval representatives
• We will express a variety of quantities in terms
  of the Riemann sum

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This illustrates that the size of ?x is allowed
to vary
y f (x)
x1 x2 x3 x4 x5
a x1 x2 x3 x4 x5
Etc
Then a lt x1 lt x2 lt x3 lt x4 .etc. is a
partition of a, b Notice the partition ?x
does not have to be the same size for each
rectangle.
And x1 , x2 , x3 , etc are x
coordinates such that a lt
x1 lt x1, x1 lt x2 lt x2 , x2 lt x3 lt x3 ,
and are used to construct the height of the
rectangles.
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The graph of a typical continuous function y
ƒ(x) over a, b. Partition a, b into n
subintervals a lt x1 lt x2 ltxn lt b. Select any
number in each subinterval ck. Form the product
f(ck)?xk. Then take the sum of these products.
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• This is called the Riemann Sum of the partition
  of ?x.
• The width of the largest subinterval of a
  partition ? is the norm of the partition, written
  x.
• As the number of partitions, n, gets larger and
  larger, the norm gets smaller and smaller.
• As n??, x ?0 only if x are the same
  width!!!!

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The Riemann SumCalculated

• Consider the function2x2 7x 5
• Use ?x 0.1
• Let the left edgeof each subinterval
• Note the sum

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The Riemann Sum

• We have summed a series of boxes
• If the ?x were smaller, we would have gotten a
  better approximation

f(x) 2x2 7x 5
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Finer partitions of a, b create more rectangles
with shorter bases.
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The Definite Integral

• The definite integral is the limit of the Riemann
  sum
• We say that f is integrable when
• the number I can be approximated as accurate as
  needed by making ? sufficiently small
• f must exist on a,b and the Riemann sum must
  exist
• is the same as saying n ?

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Notation for the definite integral
upper limit of integration
Integration Symbol
integrand
variable of integration
lower limit of integration
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Important for AP test and mine too !!
Recognizing a Riemann Sum as a Definite
integral
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Recognizing a Riemann Sum as a Definite
integral
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Recognizing a Riemann Sum as a Definite
integral
From our textbook
Notice the text uses ? instead of ?x, but
it is basically the same as our ?x , and ci
is our xi
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Try the reverse write the integral as a
Riemann Sum also on AP and my test
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Theorem 4.4 Continuity Implies Integrability
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Relationship between Differentiability,
Continuity, and Integrability
D
C
I
D differentiable functions, strongest condition
all Diff ble functions are continuous and
integrable. C continuous functions , all cont
functions are integrable, but not all are diff
ble. I integrable functions, weakest condition
it is possible they are not con t, and not
diff ble.
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Evaluate the following Definite Integral
First remember these sums and definitions
ci a i ?x
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ci a i ?x
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EXAMPLE Evaluate the definite integral
by the limit definition
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Evaluate the definite integral by the limit
definition, continued
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The Definite integral above represents the Area
of the region under the curve y f ( x) ,
bounded by the x-axis, and the vertical lines
x a, and x b
y f ( x)
y
x
a
b
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Theorem 4.4 Continuity Implies Integrability
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Relationship between Differentiability,
Continuity, and Integrability
D
C
I
D differentiable functions, strongest condition
all Diff ble functions are continuous and
integrable. C continuous functions , all cont
functions are integrable, but not all are diff
ble. I integrable functions, weakest condition
it is possible they are not con t, and not
diff ble.
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Areas of common geometric shapes
Y x
y
x
3
0
Soln to definite integral
A ½ base height
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A Sight Integral ... An integral you should know
on sight
a
-a
This is the Area of a semi-circle of radius a
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Special Definite Integrals for f
(x ) integrable from a to b
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EXAMPLE
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Additive property of integrals
y
x
c
a
b
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More Properties of Integrals
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EXAMPLE
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Even Odd Property of Integrals
Even function f ( x ) f ( - x )
symmetric about y - axis
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Finally . Inequality Properties
END
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Rules for definite integrals
Example 2
Evaluate the using the following values
60 2(2) 64
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Using the TI 83/84 to check your answers

• Find the area under on 1,5
• Graph f(x)
• Press 2nd CALC 7
• Enter lower limit 1
• Press ENTER
• Enter upper limit 5
• Press ENTER.

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Set up a Definite Integral for finding the area
of the shaded region. Then use geometry to find
the area.
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Use the limit definition to find
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Set up a Definite Integral for finding the area
of the shaded region. Then use geometry to find
the area.
rectangle
triangle