11.5 Area2014
After this lesson, you should be able to:
Use sigma notation to write and evaluate a sum.Understand the concept of area.Approximate the area of a plane region.Find the area of a plane region using limits.
ReviewExample : Evaluate the following limit:
2
1
1lim 1
n
ni
i
n n
7
3
Area
wlA hbA 2
12rA
2
2)( xxf
Area of the region bounded by and the lines x=2 and y=0?
2)( xxf
x
y
Lower ApproximationUsing 4 inscribed rectangles of equal width
Lower approximation =(sum of the rectangles)
4
91
4
10
4
2
4
14
2
14
7
2
2)( xxf
x
y
Using 4 circumscribed rectangles of equal width
Upper approximation =(sum of the rectangles)
4
4
91
4
1
4
2
4
30
2
1
4
15
2
2)( xxf
Upper Approximation
Continued…
4
91
4
10
2
1
4
14
2
1
4
7
L
4
4
91
4
1
2
1
4
30
2
1
4
15
U
L A U
4
7 A 4
15The average of the lower and upper approximations is
2
LU
2
415
47
2
422
4
11
A is approximately 4
11
The procedure we just used can be generalized to the methodology to calculate the area of a plane region. We begin with subdividing the interval [a, b] into n subintervals, each of equal width x = (b – a)/n.
Theorem 4.3 Limits of the Upper and Lower Sums
a b
n
abx
Area =
n
n
i n
abi
n
abaf
1
lim
height x base
In General - Finding Area Using the Limit
Or, xi , the i-th right endpoint
x
y
2
2)( xxf
length = 2 – 0 = 2
xnn
202
n = # of rectangles
A
n
n
i nin
f1
22 lim
n
n
i
inn 1
22
42 lim
n
n
i
inn 1
22
42 lim
Exact Area Using the Limit
in
M i
2
A
n
n
i nin
f1
22 lim
n
n
i
inn 1
22
42 lim
n
n
i
inn 1
22
42 lim
n
nnn
nn 6
)12)(1(42 lim
2
n n
n
n
n 121
3
4 lim
n nn
12
11
3
4 lim
)2)(1(3
4
3
8
Exact Area Using the Limit
Definition of the Area of a Region in the Plane
Regular Right-Endpoint Formula
RR-EFExample 6 Find the area under the graph of 2( ) 4 6 on the interval [1, 5]f x x x
1 5
n
ab
nn
415
in
aba i
n
41
A =
n
n
i nin
f1
441 lim
n
n
i
in
inn 1
2
64
144
14
lim
n
n
i
in
in
inn 1
22
616
4168
14
lim
n
n
i nin
fA1
441 lim
n
n
i
in
inn 1
2
64
144
14
lim
n
n
i
in
in
inn 1
22
616
4168
14
lim
n
n
i
in
inn 1
22
38164
lim
nn
nn
n
nnn
nn3
2
)1(8
6
)12)(1(164 lim
2
Regular Right-Endpoint Formula
nn
nn
n
nnn
nn3
2
)1(8
6
)12)(1(164 lim
2
n n
nnn
n12
)1(16)12)(1(
3
32 lim
2
n n
n
n
n
n
n12
116
121
3
32 lim
12)1(16)2)(1(3
32
n nnn12
1116
12
11
3
32 lim
Continued
3
52
Regular Right-Endpoint Formula
RR-EFExample 7 Find the area bounded by the graph of f(x),
the x-axis, the y-axis, and x = 3.
2( ) 9 on the interval [0, 3]f x x
18
Regular Right-Endpoint Formula
RR-EFExample 8 Find the area bounded by the graph of f(x),
and the x-axis on the given interval
2( ) 3 4 on the interval [1, 4]f x x x
21
2
HomeworkDay 1: Section 11.5 pg. 788 1-5 odd, 15-29 odd
Day 2: Section 11.5 pg. 788 16-30 even
Day 3: Ch. 11 Review pg. 791 3-91 odd
Day 4: Ch. 11 Practice Test
Ch. 11 Test Monday 5/11
HWQ
Find the area between the graph of f(x) and the x-axis on the given interval:
2( ) 2 on the interval [0, 1]f x x
Top Related