4-2: Area
Objectives:
Assignment:
1. To use sigma notation to • P. 267-270: 1-4, 7-9, 15,
write and evaluate a sum
19, 77
2. To find the area of a
plane region
• P. 268-270: 25-30, 47-53
odd, 78
Warm Up 1
Find the sum of the integers from 1 to 10.
1 2
+ 10 9
3
8
4
7
Karl Friedrich Gauss
5
6
11 11 11 11 11
= 5 ∙ 11
= 55
10
𝑖 = 55
𝑖=1
I did this when I
was a wee lad!
Warm Up 2
Find the sum of the integers from 1 to 100.
⋯
⋯
49
52
50
51
101 101 101 101 101 ⋯
101
101 = 50 ∙ 101
1
2
3
+ 100 99 98
4
5
97 96
= 5050
100
𝑖 = 5050
𝑖=1
Warm Up 𝑛
Find the sum of the integers from 1 to 𝑛.
𝑛
2
2
3
𝑛−1 𝑛−2
⋯
⋯
𝑛
2−1
𝑛
+2
2
𝑛
2
𝑛+1 𝑛+1 𝑛+1
⋯
𝑛+1
𝑛 + 1 = 𝑛2 ∙ 𝑛 + 1
1
+ 𝑛
+1
𝑛 𝑛+1
=
2
𝑛
𝑖=
𝑖=1
𝑛 𝑛+1
2
Objective 1
You will be able to use
sigma notation to write
and evaluate a sum
Sigma Notation
The sum of the 𝑛 terms 𝑎1 , 𝑎2 , 𝑎3 , …, 𝑎𝑛 is written
Upper bound of summation
𝑛
Index of
summation
𝑎𝑖 = 𝑎1 + 𝑎2 + 𝑎3 + ⋯ + 𝑎𝑛
𝑖=1
𝑖th term of summation
Lower bound of summation
Exercise 1
Expand the following sums.
6
1.
𝑖
𝑖=1
5
𝑖+1
2.
𝑖=0
Exercise 1
Expand the following sums.
𝑛
3.
𝑘=1
1 2
𝑘 +1
𝑛
𝑛
𝑓 𝑥𝑖 ∆𝑥
4.
𝑖=1
Properties of Summation
𝑛
𝑛
𝑘 ∙ 𝑎𝑖 = 𝑘
𝑖=1
𝑛
𝑎𝑖
𝑖=1
Constants can be taken out
𝑛
𝑎𝑖 ± 𝑏𝑖 =
𝑖=1
𝑛
𝑎𝑖 ±
𝑖=1
𝑏𝑖
𝑖=1
Sum of a sum or difference is the
sum or difference of the sums
Sum Formulas
𝑛
𝑛
𝑖2
𝑐 =𝑐∙𝑛
𝑖=1
𝑛
𝑖=1
𝑛 𝑛+1
𝑖=
2
𝑖=1
𝑛 𝑛 + 1 2𝑛 + 1
=
6
𝑛
𝑖=1
2 𝑛+1
𝑛
𝑖3 =
4
2
Exercise 2
Evaluate
10,000.
𝑛
10
100
1,000
10,000
𝑛 𝑖+1
𝑖=1 𝑛2
for 𝑛 = 10, 100, 1,000, and
𝚺
0.65
0.515
0.5015
0.50015
Approaches 1/2
𝑛+3 1
lim
=
𝑛→∞ 2𝑛
2
Objective 2
You will be able to find the
area of a plane region
The Area Problem
Recall that a classic
problem in
mathematics involved
finding the area under
a nonlinear curve.
This problem is known
as the Area Problem,
and solving it lead to
the development of
Integral Calculus.
5
4
3
2
1
2
A
–1.00
2
1
2
3
4
B
5.00
6
The Area Problem
One way to deal with
this problem is to
approximate the
unknown area by
breaking it up into a
series of rectangles.
5
4
3
2
1
2
A
–1.00
Now we try to make the width of the
rectangle as close to zero as
possible.
2
1
2
3
4
B
5.00
6
The Area Problem
One way to deal with
this problem is to
approximate the
unknown area by
breaking it up into a
series of rectangles.
5
4
3
2
1
2
A
–1.00
As the width of each rectangle
approaches 0, the sum of the areas
of the rectangles approaches the
actual area of the unknown region.
2
1
2
3
4
B
5.00
6
Exhaustion
This method of breaking a region into smaller and
smaller bits is called exhaustion, and it is related to
Archimedes’ Method of approximating 𝜋…
PiCircumscribed = 3.46410
Pi = 3.14159
PiInscribed = 3.00000
PiCircumscribed = 3.21539
Pi = 3.14159
PiInscribed = 3.10583
PiCircumscribed = 3.15966
Pi = 3.14159
PiInscribed = 3.13263
Exhaustion
…and deriving various area formulas.
Exercise 3
Use 5 rectangles to find 2 approximations of the
area of the region lying between the graph of
𝑓 𝑥 = −𝑥 2 + 5 and the 𝑥-axis between 𝑥 = 0 and
𝑥 = 2.
Rectangle Width:
2−0 2
=
5
5
Rectangle Height:
2
𝑓(𝑥) evaluated at
→𝑓 𝑖
each right endpoint
5
𝑓
2
5
,𝑓
4
5
,𝑓
6
5
,𝑓
8
5
,𝑓
10
5
Exercise 3
Use 5 rectangles to find 2 approximations of the
area of the region lying between the graph of
𝑓 𝑥 = −𝑥 2 + 5 and the 𝑥-axis between 𝑥 = 0 and
𝑥 = 2.
Rectangle Width:
2−0 2
=
5
5
Rectangle Height:
2
𝑓(𝑥) evaluated at
→𝑓
𝑖−1
each left endpoint
5
𝑓 0 ,𝑓
2
5
,𝑓
4
5
,𝑓
6
5
,𝑓
8
5
A Better Approximation
Let 𝐴 equal the area of the region region lying
between the graph of 𝑓 𝑥 = −𝑥 2 + 5 and the 𝑥axis between 𝑥 = 0 and 𝑥 = 2.
Lower Sum = 6.48 < 𝐴 < Upper Sum = 8.08
A Better Approximation
Let 𝐴 equal the area of the region region lying
between the graph of 𝑓 𝑥 = −𝑥 2 + 5 and the 𝑥axis between 𝑥 = 0 and 𝑥 = 2.
How could
we get a
better
estimate of
𝐴?
Inscribed Rectangle
<
Circumscribed Rectangle
Area of a Plane Region
Let a plane region
be defined by the
area bounded by
the graph of a
nonnegative,
continuous function
𝑦 = 𝑓(𝑥) and the 𝑥axis on the interval
𝑎, 𝑏 .
Area of a Plane Region
To find the area of the
plane region, we
divide it into 𝑛
rectangular
subintervals.
Width of Subintervals:
𝑏−𝑎
∆𝑥 =
𝑛
Endpoints of
Subintervals:
Area of a Plane Region
The height of each
rectangle can be any
function value within
the subinterval. By
EVT, each subinterval
has a minimum and a
maximum.
𝑓 𝑚𝑖 = Minimum value of 𝑓 𝑥
in the 𝑖th subinterval
𝑓 𝑀𝑖 = Maximum value of 𝑓 𝑥
in the 𝑖th subinterval
Area of a Plane Region
The height of each
rectangle can be any
function value within
the subinterval. By
EVT, each subinterval
has a minimum and a
maximum.
𝑓 𝑚𝑖 = Height of inscribed rectangle
𝑓 𝑀𝑖 = Height of circumscribed rectangle
Area of a Plane Region
The sum of the inscribed
rectangles is called the
lower sum.
The sum of the
circumscribed rectangles
is called the upper sum.
Area of a Plane Region
The sum of the inscribed
rectangles is called the
lower sum.
The sum of the
circumscribed rectangles
is called the upper sum.
𝑛
𝑠 𝑛 =
𝑓 𝑚𝑖 ∆𝑥
𝑖=1
𝑛
𝑆 𝑛 =
𝑓 𝑀𝑖 ∆𝑥
𝑖=1
Area of a Plane Region
Exercise 4
Find the lower and upper sums for the region
bounded by the graph of 𝑓(𝑥) = 𝑥 2 and the 𝑥-axis
on the interval 0,2 .
𝒏
𝟏𝟎
𝟏𝟎𝟎
𝟏𝟎𝟎𝟎
𝑠(𝑛)
2.280
2.627
2.663
𝑆(𝑛)
3.080
2.707
2.671
Limits of Lower & Upper Sums
Let 𝑓 be continuous and nonnegative on 𝑎, 𝑏 . The
limits as 𝑛 → ∞ of both lower and upper sums exist and
are equal to each other.
𝑛
lim 𝑠 𝑛 = lim
𝑛→∞
𝑛→∞
𝑏−𝑎
𝑓 𝑚𝑖 ∆𝑥
𝑖=1
𝑛
= lim
𝑛→∞
𝑓 𝑀𝑖 ∆𝑥
𝑖=1
= lim 𝑆 𝑛
𝑛→∞
Where ∆𝑥 = 𝑛 and 𝑓 𝑚𝑖 and
𝑓 𝑀𝑖 are the minimum and
maximum values of 𝑓 on the
subinterval.
Moreover, by the
Squeeze Theorem,
we can use any
arbitrary 𝑥-value in
the subinterval.
Doesn’t matter if we use the
min or max for the sum.
Left endpoint?
Right endpoint?
Exercise 5
Find the area of the
region bounded by
𝑓(𝑥) = 𝑥 3 , the
𝑥-axis, and the
vertical lines 𝑥 = 0
and 𝑥 = 1.
Exercise 6
Find the area of the
region bounded by
𝑓(𝑥) = 4 − 𝑥 2 , the
𝑥-axis, and the
vertical lines 𝑥 = 1
and 𝑥 = 2.
4-2: Area
Objectives:
Assignment:
1. To use sigma notation to • P. 267-270: 1-4, 7-9, 15,
write and evaluate a sum
19, 77
2. To find the area of a
plane region
• P. 268-270: 25-30, 47-53
odd, 78