Properties of Definite Integrals in Mathematics II Second Semester
M
A
T
H
EM
A
T
I
C
S
II
SECONDSEMESTER
The
Definite
Integral
06
T
h
e
D
e
f
i
n
i
t
e
I
n
t
e
g
r
a
l
P
r
o
pert
i
e
s
o
f
D
e
f
i
n
i
t
e
I
n
t
eg
r
al
s
When
ƒand
g
are
integrable
on
the
interval
[a
,
b],
the
definite
integral
satisfies
Rules
1
to
7
in
Table
5.3.
EX
AM
P
L
E
1
:
Using
the
Rules
for
Definite
Integrals,
Suppose
that.
EX
A
MP
L
E
2
:
Finding
Bounds
for
an
Integral
S
h
o
w
t
h
a
t
the
v
alue
o
f
is less
than
3\
2
A
r
e
a
Und
e
r
a
Cu
r
v
e
a
s
a
D
e
f
i
n
i
t
e
I
n
t
e
g
r
a
l
In
c
o
n
c
lusi
o
n
,
w
e
h
a
v
e
the
f
o
ll
o
w
in
g
rule
f
o
r
i
nt
e
g
r
a
t
in
g
f
(
x
)
=
x
.
This
formula
gives
the
area
of
a
trapezoid
d
o
w
n
t
o
the line
y
=
x
(se
e
Figu
r
e
)
.
o
n
[-
2,
2
].
So
luti
o
n
:
E
XAM
PL
E
4:
EXAMPEL
5:
EXE
R
CI
S
E
S
5
.
3
1.
Using
Properties
and
Known
Values
to
Find
Other
Integrals.
A.
B.
2
.
U
s
e
the rul
e
s
in
T
a
b
l
e
5
.
3
and
E
qu
a
ti
o
n
s
(
1
)
t
o
e
v
a
lu
a
t
e
the
integrals
in
Exercises
41–50.
3.
In
Exercises
51–54 use
a
definite
integral
to
find
the
area
of
the
region
between
the
given
curve
and the
x
-axis
on
the
interval
[0,
b
].
4.
In
Exercises
55–62,
graph
the
function
and
find
its
average
value
over
the
given
interval.
5
.
T
he
o
r
y
a
n
d
Examples
Delve into the world of definite integrals in Mathematics II Second Semester, understanding rules, properties, and applications through examples. Learn to find bounds, evaluate integrals, calculate areas under curves, and graph functions for a comprehensive understanding of this mathematical concept.
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Presentation Transcript
MATHEMATICSII SECONDSEMESTER 06 The Definite Integral
PropertiesofDefiniteIntegrals When andg are integrable on the interval [a , b], the definite integral satisfies Rules 1 to7 in Table5.3.
Usingthe Rules for Definite Integrals,Supposethat. EXAMPLE1:
EXAMPLE2: Finding Bounds for anIntegral Show that the value of is less than 3\2
In conclusion, we have the following rule for integratingf(x)= x. This formulagivesthe areaof a trapezoid down tothe line y= x(seeFigure).
EXAMPLE4: on[-2,2]. Solution :
EXERCISES5.3 1. UsingPropertiesand KnownValuestoFind Other Integrals. A. B.
2. Use the rules in Table5.3 and Equations (1) to evaluate the integralsin Exercises41 50. 3. In Exercises 51 54 use a definite integral to find the area of the region between the given curve and the x-axison the interval [0, b].
4. In Exercises5562,graph the functionand find its averagevalue over the given interval. 5. Theory and Examples
