Graphics Reference
In-Depth Information
f
or
f
exist?
Comparethis rsult with qn 33.
1
n
1
1
1
2
1
3
...
1
n
68
Find thelimit of
as
n
.
Useqn 5.59.
Summary
-
The Fundamental Theorem of Calculus
Theorem
qn 53
If
f
:[
a
,
b
]
R is integrable, then the function
F
defined by
F
(
x
)
f
, is continuous on [
a
,
b
].
Theorem
qn54
If
f
:[
a
,
b
]
R is integrable and
F
(
x
)
f
(
x
),
then
f
F
(
b
)
F
(
a
).
The Fundamental Theorem of Calculus
qn 57
If
f
:[
a
,
b
]
R is continuous and
F
(
x
)
f
,
then
F
(
x
)
f
(
x
).
Improper integrals - definitions
qn 63
If
I
(
b
)
f
and
I
(
b
)
L
as
b
, then we
write
f
L
.
qn 67
If
I
(
b
)
L
as
b
c
, then we write
f
L
.
L
as
a
qn 64
If
I
(
a
)
f
and
I
(
a
)
, then we
write
f
L
.
qn 65
If
I
(
a
)
L
as
a
c
, then we write
f
L
.
Historical note
In Archimedes' topic on the
Quadrature of the Parabola
(c. 250 ..)
he finds the area bounded by a parabolic segment by inscribing a
succession of triangles and summing their areas with a geometric series.
In this later work
On Spirals
he included a summation of segments of
areas which led to the same algebraic calculations and argument as in
our qn 1. Cavalieri, a pupil of Galileo, published further studies of area
