MATH0201 BASIC CALCULUS
MATH0201
BASIC CALCULUS
Definite Integration
Dr. WONG Chi Wing
Department of Mathematics, HKU
MATH0201 BASIC CALCULUS
Riemann Sum and Definite Integral
Evaluation of Definite Integrals
Fundamental Theorem of Calculus
Definite Integral by the Method of Substitution
Applications of Definite Integral
Area between Curves
Volume of Solids of Revolution
Applications in Economics
Applications in Life and Social Sciences
Reference
§ 5.3–5.6 of the textbook.
MATH0201 BASIC CALCULUS
Riemann Sum and Definite Integral
Archimedes’ Quadrature of Parabola
Archimedes (3rd century B.C.) found the area of a segment of a
parabola by the
method of exhaustion
.
I
BM
=
CM
and
AM
is
parallel to the axis of
parabola.
I
Repeat the process on
every subsegment of the
parabola.
It can be proved that
Area
(
AGB
) +
Area
(
AHC
) =
1
4
Area
(
ABC
)
:
The area of the segment of the parabola is
4
3
Area
(
ABC
)
.
MATH0201 BASIC CALCULUS
Riemann Sum and Definite Integral
Quadrature of Parabola in 17th Century
Consider
f
(
x
) =
x
2
over
[
0
;
1
]
.
I
Finer partition may yield a
better approximation of the
area under the curve.
I
If we partition
[
0
;
1
]
into
n
equal parts, the sum of the
area of rectangles is
(
n
1
)
n
(
2
n
1
)
6
n
3
:
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MATH0201 BASIC CALCULUS
Riemann Sum and Definite Integral
Definition 2 (Riemann Sum (p.401))
Let
f
be a function and
[
a
;
b
]
dom
(
f
)
.
I
Partition
[
a
;
b
]
into
n
subintervals:
[
x
0
;
x
1
]
,
[
x
1
;
x
2
]
,
: : :
,
[
x
n
1
;
x
n
]
where
x
0
=
a
and
x
n
=
b
.
I
The length of the subintervals may not be the same.
I
Over each subinterval
[
x
k
1
;
x
k
]
, a rectangle of height
f
(
c
k
)
is erected where
c
k
2
[
x
k
1
;
x
k
]
.
The sum of the areas of these
n
rectangles is called a
Riemann
sum of f on
[
a
;
b
]
.
MATH0201 BASIC CALCULUS
Riemann Sum and Definite Integral
Theorem 3 (Limit of Riemann Sum)
If f is continuous on
[
a
;
b
]
, then the Riemann sums of f on
[
a
;
b
]
will get closer and closer to a certain real number L by
considering finer and finer partitions.
Definition 4 (Definite Integral (p.401))
Let
f
be continuous on
[
a
;
b
]
. The limit
L
of Riemann sums of
f
over
[
a
;
b
]
is called the
definite integral of f from a to b
and it is
denoted by
Z
b
a
f
(
x
)
dx
:
f
(
x
)
is the
integrand
,
a
is the
lower limit of integration
, and
b
is
the
upper limit of integration
.
MATH0201 BASIC CALCULUS
Riemann Sum and Definite Integral
Geometric Interpretation of Definite Integral (p.399)
The definite integral
Z
b
a
f
(
x
)
dx
represents the cumulative sum
of the
signed
areas between the graph of
f
and the
x
–axis
from
x
=
a
to
x
=
b
.
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 Spring '11
 C.WONG
 Calculus, Fundamental Theorem Of Calculus, Integrals, Riemann sum

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