4.2 Area
4.2 Area
Sigma Notation
n
iia
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
The sum of n terms a1, a2, a3, ….an is written
6
1i
i
5
0
)1(i
i
7
3
2
j
j
Examples:
n
ii
n
ii akka
11
n
ii
n
ii
n
iii baba
111
)(
Properties of Summations
cncn
i
1 2
)1(
1
nni
n
i
6
12)1(
1
2
nnni
n
i 4
)1( 22
1
3
nni
n
i
Summation Formulas
Example 1: Find the sum of the first 100 integers.
Example 2: Summation Practice
5
2
31i
ii
30
1
2i
i
20
1
21j
j
Example 3: Limits Review
nnnnn
233
323
4lim
2
118lim
2
nn
nn
Example 4: Limit of a Sequence
n
in n
i
12
16lim
n
in
in1
2
31
1lim
n
in
in1
2
21
1lim
Warm-up
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
Find the area under the curve of
5)( 2 xxf
x
y
Between x = 0 and x = 2
Area of a Plane Region—Upper and Lower Sums
n
abx
bxnaxaxaxa ...210
intervalith on the maximum)(
intervalith on the minimum)(
i
i
Mf
mf
Begin by subdividing the interval [a,b] into n subintervals, each of length
Endpoints of the subintervals:
Because f is continuous, the Extreme Value Theorem guarantees the existence of a min and a max on the interval.
Rectangle Rectangle
bedCircumscri of Area)()( Inscribed of Area xMfxmf ii
Rectangle Rectangle
bedCircumscri of Area)()( Inscribed of Area xMfxmf ii
Sum of these areas=
lower sum
Sum of these areas=
upper sum
n
ii
n
ii
xMfnS
xmfns
1
1
)(SumUpper
)(SumLower
x
y
x
y
Example: Find the upper and lower sums for the region bounded by the graph
of 2 and 0between axis theand )( 2 xxxxxf
other.each toequal are sumsupper andlower both the of
as limits The b].[a, interval on the enonnegativ and continuous be fLet
Sums Upper andLower theofLimit :Theorem
n
n
abx
xcxxcf
bxax
iii
n
ii
n
where
,)(limArea
is and lines vertical theand axis,- x thef, ofgraph by the bounded
region theof area The b].[a, interval on the enonnegativ and continuous be fLet
Plane in theRegion of Area theof Definition
11
Example: Find the area of the region bounded by the graph
of 1 and 0 lines vertical theand )( 3 xxxxf
Example: Find the area of the region bounded by the graph
of 3 and 1 lines vertical theand 9)( 2 xxxxf
Example: Find the area of the region bounded by the graph
of 1 and 0 lines horizontal theand )( 2 yyyyf
x
y