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Advanced Mathematics D
Chapter Six
Intergration
The Area Problem
Given a function f that is continuous and
nonnegative on an interval [a,b], find the
area between the graph of f and the
interval [a,b] on the x-axis
Antiderivative
Definition
A function F is called an antiderivative of a
function f on a given interval I if F’(x) =f(x)
for all x in the interval.
Antiderivatiation & Integration
 If
d
[ F ( x)]  f ( x)
dx
 Then

 Or
f ( x)dx  F ( x)  C
d 
  f ( x)
f
(
x
)
dx



dx
Indefinite Integral
 Express  f ( x)dx
is called indefinite integral.
 It emphasizes that the result of
antidifferentiation is “generic” function with a
indefinite constant term
 “  “ is called an integral sign
 “f(x)” is called integrand
 “C” is called the constant of integration.
 dx is the differential symbol, serves to identify
the independent variable
Properties of the Indefinite Integral
Theorem
Suppose that F(x), G(x) are antiderivatives of
f(x),g(x), respectively, and c is a constant. Then
A constant factor can be moved through an integral sign:
 cf ( x)dx  cF ( x)  C
An antiderivation of a sum (difference) is the sum
(difference) of the antiderivatioves:
 f ( x)  g ( x)  dx  F ( x)  G( x)  C


Integration Formula

 sin xdx   cos x  C ,  cos xdx  sin x  C ,
 sec xdx  tan x  C ,  csc xdx   cot x  C ,
 sec x tan xdx  sec x  C ,  csc x cot xdx   csc x  C,
 e dx  e  C  b dx  b / ln b  C (b  0, b  1),
 (1  x ) dx  tan x  C ,  x dx  ln | x | C
dx  x  C ,
r
r 1
x
dx

x
/(r  1)  C (r  1),

2
2
x

x
x
2 1
1
dx
1 x
1
2
 sin x  C ,
x
1
x
dx
x2 1
 sec 1 | x | C.
Integration by Substitution
 Step 1 Look for some composition f(g(x)) within
the integrand for which the substitution
u  g ( x),
du  g '( x)dx
Produces an integral that is expressed entirely
in terms of u and du, (may not OK)
 Step 2 If OK in Step 1, then try to evaluate the
resolution integral in terms of u, (may not OK)
 Step 3 If OK in Step 2, then replace u by g(x) to
express your final answer in term of x.
Difinite Integral – Riemann Sum
 A function f is said to be integrable on a finite closed
n
interval [a,b] if the limit
lim  f ( xk* )xk
max xk  0
k 1
exists and does not depend on the choice of partitions or
on the choice of the points in the subintervals. When
this is the case we denote the limit by the symbol

b
a
f ( x)dx  lim
max xk  0
n

k 1
f ( xk* )xk
which is called the definite integral of f from a to b. The
numbers a and are called the lower and upper limit of
integration respectively and f(x) is called the integrand.
Difinite Integral – Graph Area
Theorem
If a function f is continuous on an interval
[a,b], then f is integrable on [a,b], and net
signed area A between the graph of f and
the interval [a,b] is
b
A   f ( x)dx
a
Definite Integral - Basic Properties
Definition
If a is in the domain of f, we define

a
a
f ( x)dx  0
If f is integrable on [a,b], then we define

b
a
a
f ( x)dx   f ( x)dx
b
Theorem
 If f and g are integrable on [a,b] and if c is a
constant, then cf, f±g are integrable on [a,b]
and
b
b



  f ( x)  g ( x) dx  
a
cf ( x)dx c  f ( x )dx
a
b
b
a
a
b
f ( x )dx   g ( x )dx
a
Theorem
 If f and g are integrable on a closed interval
containing the three points a,b and c, then


b
a
c
b
a
c
f ( x)dx  f ( x)dx   f ( x)dx
Theorem
 If f and g are integrable on [a,b] and f(x)≥0 for
all x in [a,b], then

b
a
f ( x)dx  0
 If f and g are intgraable on [a,b] and f(x) ≥ g(x)
for all x in [a,b], then

b
a
b
f ( x)dx  g ( x)dx
a
Definition
 A function f that is defined on an interval I is
said to be bounded on I if there is a positive
number M such that
 M  f ( x)  M
for all x in the interval I. Geometrically, this
means that the graph of f over the interval I lied
between the lines y=-M and y=M
Theorem
Let f be a function that is defined on the
finite closed interval [a,b]
 If f(x) has finitely many discontinuities in [a,b]
but bounded on [a,b], then f is integrable on [a,b]
If f is not bounded on [a,b], then f(x) is not
integrable on [a,b]
The Fundamental Theorem of Calculus
Part 1
Theorem
If f is continuous on [a,b] and F is any
antiderivative of f on [a,b], then

b
a
f ( x)dx F (b)  F (a)
Relationship between Indefinite and
definite Integration

b
a
f ( x)dx   F ( x)  C a
b
  F (b)  C    F (a)  C 
 F (b)  F (a)
Dummy Variable
Variable of integration in a definite integral
plays no role in the end result
So you can change the variable of the
integration whenever you feel it convenient
The Mean-Value Theorem for Integral
If f is continuous on a closed interval [a,b],
then there is at least one point x* in [a,b]
such that

b
a
f ( x)dx  f ( x )(b  a)
*
The Fundamental Theorem of Calculus
Part 2
If f is continuous on an interval I, then f
has an antiderivative on I. In particular if
a is any point in I, then the function F
defined by
x
F ( x)   f (t )dt
a
is an antiderivative of f on I; that is
d x
F '( x)  f ( x) or
f ( x)dx  f ( x).

a
dx
Integration A Rate of Change
Integrating the rate of change of F(x) with
respect to x over an interval [a,b] products
the change in the value of F(x) that
occurs as x increases from a to b.

b
a
F '( x)dx  F (a)  F (b)
Evaluation Definite Integrals by
Substitution
Method1
Calculate the indefinite integral by substitution
Using fundemantal Th. of Calculus
Method2
Direct substitute all variable including the upper
and lower limitations
Theorem
If g’ is continuous on [a,b] and f is
continuous on an interval containing
the values of g(x) for a≤x≤b, then

b
a
f ( g ( x)) g '( x)dx  
g (b )
g (a)
f (u )du
First Area Problem
Suppose that f and g are continuous
functions on an interval [a,b] and
f ( x)  g ( x)
for a  x  b
Find the area A of the region bounded
above by y= f(x), below by y= g(x), and
on the sides by the lines x=a and x=b.
Area Formula
If f and g are continuous functions on the
interval [a,b], and if f(x) ≥ g(x) for all x in
[a,b], then the area of the region bounded
above by y = f(x), below by y = g(x), on the
left by the line x=a and on the right by the
line x=b is
b
A   [ f ( x)  g ( x)]dx
a
Finding the Limits of Integraton for the
Area between Two Curves
 Step 1 Sketch the region
 Step 2 The y-coordinate of the top end point of
the line segment sketched in Step 1 will be
f(x),the bottom one g(x), and the length of the
line segment will be integrand f(x) - g(x)
 Step 3 Determine the limits. The left at which
the line segment intersects the region is x=a and
the right most is x=b.
Second Area Problem
 Suppose that w and v are continuous
functions of y on an interval [c,d] and that
w( y )  v( y )
for c  y  d
 Find the area A of the region bounded on the
left by x=v(y), right by x=w(y), and on below by
the y=c and y=d.
d
A   [w( y)  v( y)]dy
c