CHAPTER 4 SECTION 4.2 AREA

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CHAPTER 4SECTION 4.2AREA
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Sigma (summation) notation REVIEW
In this case k is the index of summation The
lower and upper bounds of summation are 1 and 5
In this case i is the index of summation The
lower and upper bounds of summation are 1 and 6
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Sigma notation
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Sigma Summation Notation
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Practice with Summation Notation
3080
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Practice with Summation Notation
Numerical Problems can be done with the TI83/84
as was done in PreCalc Algebra
Sum is in LIST, MATH Seq is on LIST, OPS
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Area Under a Curve by Limit Definition
The area under a curve can be approximated by the
sum of rectangles. The figure on the left shows
inscribed rectangles while the figure on the
right shows circumscribed rectangles
This gives the upper sum.
This gives the lower sum.
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Left endpoint approximation
(too low)
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Right endpoint approximation
(too high)
Averaging the right and left endpoint
approximations
(closer to the actual value)
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Approximating definite integralsdifferent
choices for the sample points

• If xi is chosen to be the left endpoint of the
  interval, then xi xi-1 and we have
• If xi is chosen to be the right endpoint of the
  interval, then xi xi and we have
• Ln and Rn are called the left endpoint
  approximation and right endpoint approximation ,
  respectively.

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Can also apply midpoint approximation choose
the midpoint of the subinterval as the sample
point.
The midpoint rule gives a closer approximation
than the trapezoidal rule, but in the opposite
direction.
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Midpoint rule

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Approximating the Area of a Plane Region
a.
b.
y
y
f(x) -x2 5
f(x) -x2 5
5
5
4
4
3
3
2
2
1
1
x
x
2/5 4/5 6/5 8/5 2
2/5 4/5 6/5 8/5 2
To approximate the area under each curve, you
must sum the area of each rectangle.
See next slide
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• The right endpoints, Mi, of the intervals are
  2i/5, where i 1, 2, 3, 4, 5. The width of each
  rectangle is 2/5 and the height of each rectangle
  can be obtained by evaluating f at the right
  endpoint of each interval.
• 0, 2/5, 2/5, 4/5, 4/5, 6/5, 6/5,
  8/5, 8/5, 10/5
• Evaluate f(x) at the right endpoints of each
  of these intervals.
• The sum of the area of the five rectangles is
• Height Width
• Because each of the five rectangles lies inside
  the parabolic region, you can conclude that the
  area of the parabolic region is greater than 6.48.

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Approximating the Area of a Plane Region for b
(cont)

• b. The left endpoints of the five intervals are
  2/5(i _ 1), where i 1, 2, 3,
• 4, 5. The width of each rectangle is 2/5,
  and the height of each
• rectangle can be found by evaluating f at
  the left endpoint of each
• interval.
• Height Width
• Because the parabolic region lies within the
  union of the five rectangular
• region, that the area of the parabolic region is
  less than 8.08.
• 6.48 lt Area of region lt 8.08

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ON CALCULATOR
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In general for the upper sum S(n) and Lower sum
s(n), you use the following for curves f(x) bound
between xa and xb.
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Finding Upper and Lower Sums for a Region

• Find the upper and lower sums for the region
  bounded by the graph of f(x) x2
• and the x-axis between x 0 and x 2
• Solution Begin by partitioning the interval 0,
  2 into n subintervals, each of length

B.
A.
f(x) x2
f(x) x2
4
4
3
3
2
2
1
1
1 2
1 2
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Left endpoints Right endpoints
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Limit of the Lower and Upper Sums
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Definition of the Area of a Region in the
Plane
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Area Under a Curve by Limit Definition
If the width of each of n rectangles is ?x, and
the height is the minimum value of f in the
rectangle, f(Mi), then the area is the limit of
the area of the rectangles as n? ?
This gives the lower sum.
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Area under a curve by limit definition
If the width of each of n rectangles is ?x, and
the height is the maximum value of f in the
rectangle, f(mi), then the area is the limit of
the area of the rectangles as n? ?
This gives the upper sum.
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Area under a curve by limit definition
The limit as n ? of the Upper Sum The
limit as n ? of the Lower Sum The area
under the curve between x a and x b.
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Theorem 4.3 Limits of the Lower and Upper Sums
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Definition of the Area of a Region in the Plane
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Visualization
f(ci)
ci
Width ?x
ith interval
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Example Area under a curve by limit definition
Find the area of the region bounded by the graph
f(x) 2x x3 , the x-axis, and the vertical
lines x 0 and x 1, as shown in the figure.
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Area under a curve by limit definition
Why is right, endpoint i/n?
Suppose the interval from 0 to 1 is divided into
10 subintervals, the endpoint of the first one is
1/10, endpoint of the second one is 2/10 so the
right endpoint of the ith is i/10.
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Visualization again
f(ci)
ci i/n
Width ?x
ith interval
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Find the area of the region bounded by the graph
f(x) 2x x3 on 0, 1
Sum of all the rectangles
Right endpoint
Use rules of summation
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continued
Foil Simplify
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The area of the region bounded by the graph f(x)
2x x3 , the x-axis, and the vertical lines x
0 and x 1, as shown in the figure .75
0.75
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Practice with Limits
Multiply out
Separate
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