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)
i0
7
j3
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 1i 3
i2
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
ba
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
ba
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
0
