4.2 Area
Sigma Notation
The sum of n terms a1, a2, a3, ….an is written
n
a
i
i 1
where i is the index of summation, ai is the ith
term, and the lower and upper bounds of
summation are 1 and n respectively
Examples:
6
i
i 1
5
 ( i  1)
i0
7

j3
j
2
Properties of Summations
n
 ka
n
i
i 1
 k  ai
i 1
n
 (a
n
i
 bi ) 
i 1
a
i 1
n
i

b
i
i 1
Summation Formulas
n

c  cn
i
i 1
2

n ( n  1)
i 1
2
i
i 1
n
n
n ( n  1)  2 n  1 
6
i
i 1
n ( n  1)
2
n
3

4
2
Example 1: Find the sum of the first 100 integers.
Example 2: Summation Practice
5
 i  1i  3 
i2
30
 2i
i 1
20
  j  1
j 1
2
Example 3: Limits Review
 4 
3
2
lim 
2
n

3
n
n

3
n  3n



 18
lim  2
n  n

  n n  1 


2



Example 4: Limit of a Sequence
n
lim
n 

i 1
n
lim
n 
16 i
n
1
n
i 1
2
3
i  1  2
Warm-up
n
lim
n 
1
n
i 1
2
i  1  2
Definition of the Area of a
Rectangle: A=bh
Take a rectangle whose
area is twice the triangle:
A=1/2 bh
For any polygon,
just divide the
polygon into
triangles.
Area of Inscribed Polygon < Area Circle < Area of Circumscribed Polygon
Area of a Plane Region
y

Find the area under the curve of

f ( x)   x  5
2

Between x = 0 and x = 2


x






Area of a Plane Region—Upper and Lower Sums
Begin by subdividing the interval [a,b] into
n subintervals, each of length
x 
ba
n
Endpoints of the subintervals:
a  0  x  a  1 x  a  2  x  ...  a  n  x  b
Because f is continuous, the Extreme Value Theorem guarantees the existence
of a min and a max on the interval.
f ( m i )  minimum
f ( M i )  maximum
Area of Inscribed
Rectangle
on the ith interval
on the ith interval
 f ( m i )  x  f ( M i )  x  Area of Circumscri
Rectangle
bed
 f ( m i )  x  f ( M i )  x  Area of Circumscri
Area of Inscribed
Rectangle
bed
Rectangle
Sum of these
areas=
Sum of these
areas=
lower sum
upper sum
y
Lower Sum  s ( n ) 
y

n

 f m  x
i

i 1

n
Upper Sum  S ( n ) 
 f  M  x






i
i 1
x
x












Example: Find the upper and lower sums for the region bounded by the graph
of f ( x )  x 2 and the x  axis between
x  0 and x  2
Theorem
: Limit of the Lower and Upper Sums
Let f be continuous
n   of both the
Definition
e on the interval [a, b]. The limits
and nonnegativ
in the Plane
e on the interval [a, b]. The area of the region
by the graph of f, the x - axis, and the vertical
n
Area  lim
n 
where  x 
ba
n

i 1
as
lower and upper sums are equal to each other.
of the Area of Region
Let f be continuous
bounded
and nonnegativ
f (ci )  x ,
x i 1  c i  x i
lines x  a and x  b is
Example: Find the area of the region bounded by the graph
of
f ( x )  x and the vertical lines x  0 and x  1
3
Example: Find the area of the region bounded by the graph
of
f ( x )  9  x and the vertical lines x  1 and x  3
2
Example: Find the area of the region bounded by the graph
of
f ( y )  y and the horizontal
2
y

x


lines y  0 and y  1