Transcript 4 3 Riemann

4-3: Mostly Riemann
Objectives:
Assignment:
1. To define and use
Riemann sums
• P. 277: 2
2. To define and use
definite integrals
• P. 277-280: 3-11 odd, 1720, 23-31 odd, 43, 44, 47,
49, 63-68
Warm Up
Find the area of the region bounded by
𝑓(𝑥) = 𝑥, the 𝑥-axis, and the vertical lines
𝑥 = 0 and 𝑥 = 1.
𝑛
Area = lim
𝑛→∞
= lim
𝑛→∞
𝑓
𝑖=1
1
𝑛
𝑖
𝑛
𝑛
𝑖=1
1
𝑛
𝑖
𝑛
Unfortunately, we
don’t have a formula
for the sum of the
square roots.
But what if we let
𝑐𝑖 =
𝑖2
𝑛2
…
Objective 1
You will be able to define and
use Riemann sums
Area of a Region in the Plane
Let 𝑓 be continuous and nonnegative on 𝑎, 𝑏 . The
area of the region bounded by the graph of 𝑓, the
𝑥-axis, and the vertical lines 𝑥 = 𝑎 and 𝑥 = 𝑏 is
𝑛
Area = lim
𝑛→∞
𝑓 𝑐𝑖 ∆𝑥
Riemann sum
𝑖=1
𝑥𝑖−1 ≤ 𝑐𝑖 ≤ 𝑥𝑖
𝑏−𝑎
∆𝑥 =
𝑛
Regular
partition
EK 3.2A1: A Riemann
sum, which requires a
partition of an interval 𝐼,
is the sum of products,
each of which is the value
of the function at a point
in the subinterval
multiplied by the length
of that subinterval of the
partition.
Another Notation
The value 𝑐𝑖 represents any arbitrary 𝑥-value in the
𝑖th subinterval 𝑥𝑖−1 , 𝑥𝑖 on 𝑎, 𝑏 where
𝑎 = 𝑥0 < 𝑥1 < 𝑥2 < ⋯ < 𝑥𝑖−1 < 𝑥𝑖 < ⋯ < 𝑥𝑛 = 𝑏
Another Notation
The value 𝑥𝑖∗ represents any arbitrary 𝑥-value in the
𝑖th subinterval 𝑥𝑖−1 , 𝑥𝑖 on 𝑎, 𝑏 where
𝑎 = 𝑥0 < 𝑥1 < 𝑥2 < ⋯ < 𝑥𝑖−1 < 𝑥𝑖 < ⋯ < 𝑥𝑛 = 𝑏
Width of Subinterval:
∆𝑥 =
𝑏−𝑎
𝑛
or
∆𝑥 = 𝑥𝑖 − 𝑥𝑖−1
Exercise 1
Find the area of the
region bounded by
𝑓(𝑥) = 𝑥, the 𝑥-axis,
and the vertical lines
𝑥 = 0 and 𝑥 = 1.
Use
𝑥𝑖∗
=
𝑖2
.
2
𝑛
Width of Subinterval:
∆𝑥𝑖 = 𝑥𝑖 − 𝑥𝑖−1
𝑖2
𝑖−1
∆𝑥𝑖 = 2 −
𝑛
𝑛2
2
Width of subinterval
is variable
Riemann Sum
Let 𝑓 be defined on the closed interval 𝑎, 𝑏 , and
let ∆ be a partition of 𝑎, 𝑏 given by
𝑎 = 𝑥0 < 𝑥1 < 𝑥2 < ⋯ < 𝑥𝑖−1 < 𝑥𝑖 < ⋯ < 𝑥𝑛 = 𝑏
where ∆𝑥𝑖 is the width of the 𝑖th subinterval. If 𝑥𝑖∗
represents any point in the 𝑖th subinterval, then the
sum
𝑛
𝑓 𝑥𝑖∗ ∆𝑥𝑖
𝑥𝑖−1 ≤ 𝑥𝑖∗ ≤ 𝑥𝑖
𝑖=1
is called a Riemann sum of 𝑓 for the partition ∆.
Bernhard Riemann, c. 1860
Exercise 2
Find the area of the region bounded by
𝑓 𝑥 = 𝑥 2 − 2𝑥 on the interval 0,3 .
Net Area
In a Riemann sum, 𝑓 need not be nonnegative…
If 𝑓(𝑥) ≥ 0, then a Riemann
sum can be interpreted as an
approximation of the area
under 𝑓.
If 𝑓(𝑥) ≥ 0, then the limit of a
Riemann sum can be
interpreted as the exact area
under 𝑓.
Net Area
In a Riemann sum, 𝑓 need not be nonnegative…
If 𝑓(𝑥) is not necessarily
nonnegative, then a Riemann
sum can be interpreted as an
approximation of the net area
above and below 𝑓.
If 𝑓(𝑥) is not necessarily
nonnegative, then the limit of a
Riemann sum can be
interpreted as the exact net
area above and below 𝑓.
Objective 2
You will be able to
define and use
definite integrals
Norm
The width of the largest subinterval ∆𝑥𝑖 is
called the norm, denoted as ∆ or 𝑚𝑎𝑥∆𝑥𝑖 .
The number of subintervals 𝑛 and
𝑏−𝑎
the norm ∆ are related by ∆ ≤ 𝑛.
𝑏−𝑎
∆ →0 ∆
lim
= ∞, so as ∆ → 0, 𝑛 → ∞. Therefore,
these expressions are considered equivalent.
Definite Integral
If 𝑓 is defined on the closed interval 𝑎, 𝑏 and limit
𝑛
𝑓 𝑥𝑖∗ ∆𝑥𝑖
lim
∆ →0
𝑖=1
exists, then 𝑓 is integrable on 𝑎, 𝑏 and the limit is
denoted by
Upper limit of
𝑛
𝑓 𝑥𝑖∗ ∆𝑥𝑖 =
lim
∆ →0
𝑏
𝑖=1
integration
𝑓 𝑥 𝑑𝑥 .
𝑎
Lower limit of integration
This limit is
called the
definite
integral of 𝑓
from 𝑎 to 𝑏.
Definite Integral
If 𝑓 is defined on the closed interval 𝑎, 𝑏 and limit
𝑛
𝑓 𝑥𝑖∗ ∆𝑥𝑖
lim
𝑚𝑎𝑥∆𝑥𝑖 →0
𝑖=1
exists, then 𝑓 is integrable on 𝑎, 𝑏 and the limit is
denoted by
Upper limit of
𝑛
𝑓 𝑥𝑖∗ ∆𝑥𝑖 =
lim
𝑚𝑎𝑥∆𝑥𝑖 →0
𝑏
𝑖=1
integration
𝑓 𝑥 𝑑𝑥 .
𝑎
Lower limit of integration
This limit is
called the
definite
integral of 𝑓
from 𝑎 to 𝑏.
Definite Integral
If 𝑓 is defined on the closed interval 𝑎, 𝑏 and limit
𝑛
𝑓 𝑥𝑖∗ ∆𝑥𝑖
lim
𝑛→∞
𝑖=1
exists, then 𝑓 is integrable on 𝑎, 𝑏 and the limit is
denoted by
Upper limit of
𝑛
𝑓 𝑥𝑖∗ ∆𝑥𝑖 =
lim
𝑛→∞
𝑏
𝑖=1
integration
𝑓 𝑥 𝑑𝑥 .
𝑎
Lower limit of integration
This limit is
called the
definite
integral of 𝑓
from 𝑎 to 𝑏.
Definite Versus Indefinite
Leibniz chose ∫ as the notation for an
integral since it is the limit of sums.
Definite Integral
Indefinite Integral
Number
Family of functions
Net area above/below 𝑓
Antiderivative of 𝑓′
Gottfried Leibniz, c. 1675
Exercise 1 (Revisited)
Find the area of the
region bounded by
𝑓(𝑥) = 𝑥, the 𝑥-axis,
and the vertical lines
𝑥 = 0 and 𝑥 = 1.
Rewrite as a definite integral.
𝑛
lim
𝑛→∞
𝑖=1
𝑖 2 2𝑖 − 1
𝑛2
𝑛2
Exercise 2 (Revisited)
Find the area of the
region bounded by
𝑓 𝑥 = 𝑥 2 − 2𝑥 on the
interval 0,3 .
Rewrite as a definite integral.
𝑛
lim
𝑛→∞
𝑖=1
3𝑖
𝑛
2
3𝑖
−2
𝑛
3𝑖
𝑛
Exercise 3
Evaluate
1
∫−2 2𝑥 𝑑𝑥.
Exercise 4
1
∫−2 2𝑥 𝑑𝑥.
Evaluate
Notice that the area
above the curve is a
triangle, as is the area
below the curve.
Evaluate the definite
integral geometrically.
Exercise 5
Sketch the region corresponding to each definite
integral. Then evaluate each integral using a
geometric formula.
3
1. ∫1 4 𝑑𝑥
3
2. ∫0 𝑥 + 2 𝑑𝑥
2
3. ∫−2 4 − 𝑥 2 𝑑𝑥
Special Definite Integrals
If 𝑓 is defined at 𝑥 =One
𝑎, then
𝑎
∫𝑎 𝑓(𝑥) 𝑑𝑥
= 0.
If 𝑓 is integrable on 𝑎, 𝑏 , then
Two 𝑏
𝑎
∫𝑏 𝑓(𝑥) 𝑑𝑥 = − ∫𝑎 𝑓(𝑥) 𝑑𝑥.
Exercise 6
Evaluate each definite integral.
𝜋
1. ∫𝜋 sin 𝑥 𝑑𝑥
2.
−2
∫1 2𝑥 𝑑𝑥
Additive Interval Property
If 𝑓 is integrable on 𝑎, 𝑐 ,
𝑐, 𝑏 , and 𝑎, 𝑏 , then
𝑐
𝑏
∫𝑎 𝑓(𝑥) 𝑑𝑥 + ∫𝑐 𝑓(𝑥) 𝑑𝑥
𝑏
∫𝑎 𝑓(𝑥) 𝑑𝑥.
=
Exercise 7
Use the Additive Interval
Property to evaluate
1
∫−1 𝑥 𝑑𝑥.
Properties of Definite Integrals
If 𝑓 and 𝑔 are integrable on 𝑎, 𝑏 and 𝑐 is a
constant, then the functions 𝑐𝑓 and 𝑓 ± 𝑔 are
integrable on 𝑎, 𝑏 , and
𝑏
𝑏
𝑐𝑓(𝑥) 𝑑𝑥 One
=𝑐
𝑎
𝑏
𝑓(𝑥) 𝑑𝑥
𝑎
𝑏
𝑏
𝑓(𝑥) ± 𝑔(𝑥) 𝑑𝑥 =Two 𝑓(𝑥) 𝑑𝑥 ±
𝑎
𝑎
𝑔(𝑥) 𝑑𝑥
𝑎
𝑏
𝑐 𝑑𝑥Three
=𝑐 𝑏−𝑎
𝑎
Exercise 8
Evaluate each definite integral.
1.
3 2
∫1 𝑥 𝑑𝑥
2.
3
∫1 𝑥 𝑑𝑥
3.
3
∫1 𝑑𝑥
Exercise 9
Evaluate
3
2
−𝑥
∫1
+ 4𝑥 − 3 𝑑𝑥.
Preservation of Inequality
If 𝑓 is integrable and
nonnegative on the
closed interval 𝑎, 𝑏 ,
then
𝑏
∫𝑎 𝑓(𝑥) 𝑑𝑥 ≥ 0.
If 𝑓 and g are
integrable on the
closed interval 𝑎, 𝑏
and 𝑓(𝑥) ≤ 𝑔(𝑥), then
𝑏
∫𝑎 𝑓(𝑥) 𝑑𝑥 ≤
𝑏
∫𝑎 𝑔(𝑥) 𝑑𝑥.
Preservation of Inequality
If 𝑓 is integrable and
nonnegative on the
closed interval 𝑎, 𝑏 ,
then
𝑏
∫𝑎 𝑓(𝑥) 𝑑𝑥 ≥ 0.
If 𝑓 and g are
integrable on the
closed interval 𝑎, 𝑏
and 𝑓(𝑥) ≤ 𝑔(𝑥), then
𝑏
∫𝑎 𝑓(𝑥) 𝑑𝑥 ≤
𝑏
∫𝑎 𝑔(𝑥) 𝑑𝑥.
4-3: Mostly Riemann
Objectives:
Assignment:
1. To define and use
Riemann sums
• P. 277: 2
2. To define and use
definite integrals
• P. 277-280: 3-11 odd, 1720, 23-31 odd, 43, 44, 47,
49, 63-68