MAT 1235
Calculus II
4.1, 4.2 Part I
The Definite Integral
http://myhome.spu.edu/lauw
Homework

WebAssign HW 4.2 I
Major Themes in Calculus I
The Tangent Problem
The Velocity Problem
t 2
y  f ( x)
xa
f ( a  h)  f ( a )
lim
h 0
h
Abstract World
y  f (t )
ta
lim
h 0
f ( a  h)  f ( a )
h
Real World
Major Themes in Calculus I
The Tangent Problem
y  f ( x)

xa

f ( a  h)  f ( a )
lim
h 0
h
Abstract World
We do not like to
use the definition
Develop
techniques to deal
with different
functions
Major Themes in Calculus II
The Energy Problem
The Area Problem
f ( x)
y  f ( x)
f ( x)  0 on [a, b]
y  f ( x)
Real World
Abstract World
n
A  lim  f ( xi ) x
n 
i 1
Major Themes in Calculus II
The Area Problem

y  f ( x)
f ( x)  0 on [a, b]

We do not like to
use the definition
Develop
techniques to deal
with different
functions
Abstract World
n
A  lim  f ( xi ) x
n 
i 1
Preview



Look at the definition of the definite
integral 𝑦 = 𝑓(𝑥) on [𝑎, 𝑏]
Look at its relationship with the area
between the graph 𝑦 = 𝑓(𝑥) and the 𝑥axis on [𝑎, 𝑏]
Properties of Definite Integrals
Example 0
f ( x)  x
2
on [1,5]
f ( x)  x
Example 0
2
on [1,5]
Use left hand end
points to get an
estimation
f ( 4.5)
f ( 4)
f ( 2)
f (1.5)
f (1)
Example 0
f ( x)  x
Use right hand end
points to get an
estimation
2
on [1,5]
f (5)
f ( 4.5)
f ( 2.5)
f ( 2)
f (1.5)
Example 0 Observation:
What happen to the estimation if we increase the
number of subintervals?
In General

i
f (x )
𝑖 th subinterval
sample point
x i
In General
Suppose 𝑓 is a continuous function
defined on [𝑎, 𝑏], we divide the interval
[𝑎, 𝑏] into 𝑛 subintervals of equal width
ba
x 
n
The area of the 𝑖𝑡ℎ rectangle is
f ( xi )x
In General
f ( xi )x
ith subinterval
sample point
In General
Sum of the area of the rectangles is

1

2

3

n
f ( x )x  f ( x )x  f ( x )x    f ( x )x
n
  f ( xi )x
i 1
Riemann Sum
In General
Sum of the area of the rectangles is

1

2

3

n
f ( x )x  f ( x )x  f ( x )x    f ( x )x
n
  f ( xi )x
i 1
Sigma Notation for
summation
In General
Sum of the area of the rectangles is

1

2

3

n
f ( x )x  f ( x )x  f ( x )x    f ( x )x
n
  f ( x )x
i 1
Index

i
Final value
(upper limit)
Initial value
(lower limit)
In General
Sum of the area of the rectangles is

1

2

3

n
f ( x )x  f ( x )x  f ( x )x    f ( x )x
n
  f ( xi )x
i 1
As we increase 𝑛, we get better and better
estimations.
Definition
The Definite Integral of 𝑓 from 𝑎 to 𝑏

b
a
f ( x)dx  lim
n 
n

i 1
f ( xi )x
Definition
The Definite Integral of 𝑓 from 𝑎 to 𝑏
upper limit

b
a
f ( x)dx  lim
lower limit
integrand
n 
n

i 1
f ( xi )x
Definition
The Definite Integral of 𝑓 from 𝑎 to 𝑏

b
a
f ( x)dx  lim
n 
n

i 1
f ( xi )x
Integration : Process of computing
integrals
Example 1
Express the limit as a definite integral on
the given interval.
n


lim  cos( xi )  xi x, [0,  ]
n 
i 1

b
a
f ( x)dx  lim
n 
n
 f (x
i 1
a  ?, b  ?
f ( x)  ?

i
)x
Example 1
Express the limit as a definite integral on
the given interval.
n


lim  cos( xi )  xi x, [0,  ]
n 
i 1
a  ?, b  ?
f ( x)  ?
a
b
f ( x) 
n


lim  cos( xi )  xi x 
n 
i 1

b
a
f ( x)dx  lim
n 
n
 f (x
i 1

i
)x
Remarks



We are not going to use this limit
definition to compute definite integrals.
In section 4.3, we are going to use
antiderivative (indefinite integral) to
compute definite integrals.
We will use this limit definition to derive
important properties for definite integrals.
More Remarks
If 𝑓(𝑥) ≥ 0 on [𝑎, 𝑏], then

b
a
f ( x)dx

b
a
f ( x)dx  Area " under" f
More Remarks
If 𝑓(𝑥) ≥ 0 on [𝑎, 𝑏], then
If 𝑓(𝑥) ≤ 0 on [𝑎, 𝑏], then

b
a

b
a
f ( x)dx  Area " under" f
f ( x)dx   Area " above" f
Ai   f ( xi )x

b
a

f ( x)dx
b
  f ( x)dx
a

 f ( xi )


More Remarks
n
n
Area  lim  Ai  lim    f ( xi )x 
n 
i 1
n 
i 1

Ai   f ( xi )x

b
  f ( x)dx
a

 f ( xi )


Example 2
y  f (x)
4
a

b

c
a
b
b
f ( x)dx 
f ( x)dx 
3
c
Example 3
2
Compute 1 ( x  1)dx by interpreting it in
terms of area
y  x 1
1

2
1
( x  1)dx 
1
2
Example 4
Compute

3
3
9  x 2 dx
Properties

The follow properties are labeled
according to the textbook.
Property (a)

b
a
b
f ( x)dx   f (t )dt
a
𝑥, 𝑡 are called the dummy variables
Example 5
y  x 1

2
1
( x  1)dx 
1
x
1
2
y  t 1

2
1
(t  1)dt 
1
1
t
2
Property (b)
The definition of definite integral is welldefined even if
1
upper limit < lower limit e.g. 3 f (t )dt
And

a
b
b
f ( x)dx   f ( x)dx
a
Property (b)
The definition of definite integral is welldefined even if
1
upper limit < lower limit e.g. 3 f (t )dt
And

a
b
b
f ( x)dx   f ( x)dx
a b
x 
n
a
n
lim  f ( x ) x
n 
i 1

i
ba
x 
n
Example 6
y  x 1

2
1
1
( x  1)dx 
2
1
x
1
2
1
 ( x 1)dx 
2
Note: If lower limit > upper limit, the integral has
no obvious geometric meaning
Example 7
If

3
1
f ( x)dx  4 , what is
1

3
f (t )dt ?
Example 7
If

3
1
f ( x)dx  4 , what is
1

3
f (t )dt ?
Q1: What is the answer?
Q2: How many steps are needed to clearly
demonstrate the solutions?
Property (c)

a
a
f ( x)dx  0
Example 8
1
 ( x  1)dx 
 (sin x  tan x  4
1
3
3
2
x )dx 
Classwork


2 persons per group. Work with your
partner and your partner ONLY.
Once you get checked, you can go.