1322Notes(4) - MAT1322 Notes By Dr. Hua Contents Chapter 5...

MAT1322 – Notes — By Dr. Hua
Contents
Chapter 5 – Integrals 3
5.10 Improper Integrals
. . . . . . . . . . . . . . . . . . . . . . . 3
Chapter 6 – Applications of Integration 7
6.1 More about Areas
. . . . . . . . . . . . . . . . . . . . . . . . . 7
6.2 Volume
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6.3 Volumes by Cylindrical Shells
. . . . . . . . . . . . . . . 9
6.4 Arc Length
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6.5 Average Value of a Function
. . . . . . . . . . . . . . . . 12
6.6 Applications to Physics and Engineering
. . . . . 12
Chapter 7 – Di ff erential Equations 18
7.1 Modeling with di ff erential equations
. . . . . . . . 18
7.2 Direction Fields and Euler’s method
. . . . . . . . . 18
7.3 Separation of variables
. . . . . . . . . . . . . . . . . . . . . 21
7.4 Exponential Growth and Decay
. . . . . . . . . . . . . 24
7.5 The Logistic Equation
. . . . . . . . . . . . . . . . . . . . . . 27
Chapter 8 – Infinite Sequences and Series 32
8.1 Sequences
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
8.2 Series
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8.3 The Integral and Comparison Test; Estimating
sums
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
8.4 Other Convergence Tests
. . . . . . . . . . . . . . . . . . . 37
8.5 Power Series
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
8.6 Representations of Functions as Power Series
. 42
8.7 Taylor and Maclaurin Series
. . . . . . . . . . . . . . . . 44
1

Chapter 9 – Vectors and the Geometry of Space 49
9.6 Functions and Surfaces
. . . . . . . . . . . . . . . . . . . . . 49
Chapter 11 – Partial Derivatives 50
11.1 Functions of Several Variables
. . . . . . . . . . . . . . 50
11.3 Partial Derivatives
. . . . . . . . . . . . . . . . . . . . . . . . 51
11.4 Tangent Planes and Linear Approximations
. 53
11.5 The Chain Rule
. . . . . . . . . . . . . . . . . . . . . . . . . . 55
11.6 Directional Derivatives and the Gradient Vec-
tor
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2

Chapter 5 – Integrals
5.10 Improper Integrals
Type I: Infinite Intervals
∞
a
f ( x ) dx = lim
t →∞
t
a
f ( x ) dx,
b
-∞
f ( x ) dx = lim
t →∞
b
t
f ( x ) dx,
∞
-∞
f ( x ) dx =
∞
c
f ( x ) dx +
c
-∞
f ( x ) dx.
Definition : Integral is convergent (or divergent) ⇔ Integral is a finite number (or ∞ ).
Example 1
∞
1
1
x 2
dx = 1 .
Example 2
∞
1
1
x
dx = ∞ .
Example 3
1
-∞
1
1 + x 2
dx =
3 π
4
.
Example 4
∞
-∞
e
- | x |
dx = 2 .
Example 5 Determine if the integral
I =
∞
2
xe
- x
dx
is convergent or divergent and evaluate if it is convergent.
3

Solution. I = lim
t →∞
t
2
xe
- x
dx (integration by parts: let u = x and dv = e - x dx )
= lim
t →∞
x ( - e
- x
) |
t
2
-
t
2
- e
- x
dx
= lim
t →∞
( - xe
- x
- e
- x
) |
t
2
= lim
t →∞
- ( x + 1) e
- x
|
t
2
= lim
t →∞
( - ( t + 1) e
- t
+ 3 e
- 2
)
= 3 e
- 2
(where lim
t →∞
( t + 1) e - t = 0 by L’Hospital’s Rule).
Type 2: Discontinuous Integrands
If f ( x ) is continuous on [ a, b ), then
b
a
f ( x ) dx = lim
t → b
t
a
f ( x ) dx ;
If f ( x ) is continuous on ( a, b ], then
b
a
f ( x ) dx = lim
t → a
b
a
f ( x ) dx ;
If f ( x ) is discontinuous at c : a < c < b , then
b
a
f ( x ) dx =
c
a
f ( x ) dx +
b
c
f ( x ) dx.
Example 6
3
2
1
√
3 - x
dx = lim
t → 3
t
2
1
√
3 - x
dx = lim
t → 3
[ - 2
√
3 - x ]
t
2
= 2 .
Example 7 Determine if the integral
2
0
1
x - 1
dx
is convergent or divergent and evaluate if it is convergent.
Example 8
e
0
ln xdx = 0 .
4

p -Integral
Example 9
∞
1
1
x
p
dx =
1
p - 1
, if p > 1 ;
divergent, if p ≤ 1 .
Example 10
1
0
1
x
p
dx =
1
1 - p
, if p < 1 ;
divergent, if p ≥ 1 .
Comparison Test for Improper Integral
If f ( x ) and g ( x ) are continuous and f ( x ) ≥ g ( x ) ≥ 0 on x ≥ a . Then
(i)
∞
a
f ( x ) dx is convergent = ⇒
∞
a
g ( x ) dx is convergent;
(ii)
∞
a
g ( x ) dx is divergent = ⇒
∞
a
f ( x ) dx is divergent.
Example 11
∞
1
1
√
x
3
+ 1
dx = convergent.
∵
1
√
x
3
+ 1
≤
1
x 3 / 2
.
Example 12
∞
8
1 +
√
x
x - 6
dx = divergent.