This is not easy to prove in general, but for the
Property 5 can be seen from the geometric interp
from
a
to
c
plus the area from
c
to
b
is eq
y
f x
y
c
a
f x dx
y
b
c
f x dx
4
3
y
1
0
4
3
x
2
dx
y
1
0
4
dx
y
1
0
x
y
1
0
4
dx
4 1
0
y
1
0
4
3
x
2
dx
y
1
0
4
dx
y
1
0
3
x
2
d
f
t
f
t
y
b
a
f x dx
lim
n
l
n
i
1
f x
i
lim
n
l
n
i
1
f x
y
b
a
f x
t
x
dx
lim
n
l
n
i
1
f x
f
t
374

CHAPTER 5
INTEGRALS
y
0
x
a
b
f
g
f+g
FIGURE 14
j
[
ƒ+©
]
dx=
j
ƒ
dx+
j
©
dx
a
b
a
b
a
b
N
Property 3 seems intuitively reasonable
because we know that multiplying a function
by a positive number
stretches or shrinks its
graph vertically by a factor of
. So it stretches
or shrinks each approximating rectangle by a
factor
and therefore it has the effect of
multiplying the area by .
c
c
c
c
FIGURE 15
0
y
x
a
b
c
y=ƒ
It is wellknown that
ˆ
1
0
e
x
2
dx
does not have a closedform expression. However, one can
estimate its value from above using Property (5): for any
x
in
[
0, 1
]
, we have
x
2
≤
x
and so
e
x
2
≤
e
x
. Therefore,
ˆ
1
0
e
x
2
dx
≤
ˆ
1
0
e
x
dx
=
e

1.
96
Integration
5.3
Fundamental Theorem of Calculus
In the previous sections, we talked about the indefinite and definite integrals of a given function
f
(
x
)
. The former is merely the antiderivative of
f
(
x
)
, while the latter represents the area
under the graph
y
=
f
(
x
)
. They are
a priori
different things, but the following important
theorem (Fundamental Theorem of Calculus) relates them together. Essentially, this theorem
asserts that finding area under the graph of a
continuous
function can be done by computing
antiderivative. This explains why both antiderivatives and area under a graph are called
integrals
, and that they use similar symbols.
Theorem 5.4
— Fundamental Theorem of Calculus.
Given a function
f
(
x
)
which is
continu
ous
on
[
a
,
b
]
, and
F
(
x
)
is an antiderivative of
f
(
x
)
, i.e.
F
0
(
x
) =
f
(
x
)
, then we have:
d
dx
ˆ
x
a
f
(
t
)
dt
=
f
(
x
)
and
ˆ
b
a
f
(
x
)
dx
=
F
(
b
)

F
(
a
)
i
Note that the definite integral
ˆ
x
a
f
(
t
)
dt
is a function of
x
but not
t
. Graphically, this
number represents the area under the graph
y
=
f
(
t
)
from
t
=
a
to
t
=
x
as shown below:
Isaac Barrow (1630–1677), discovered that thes
related. In fact, he realized that differentiation and
Fundamental Theorem of Calculus gives the pre
derivative and the integral. It was Newton and Leib
used it to develop calculus into a systematic mathe
that the Fundamental Theorem enabled them to c
without having to compute them as limits of sums
The first part of the Fundamental Theorem deals
of the form
where
is a continuous function on
and
v
depends only on
, which appears as the variable u
number, then the integral
is a definite num
also varies and defines a function of
den
If
happens to be a positive function, then
graph of
from
to , where
can vary from
to
tion; see Figure 1.)
EXAMPLE 1
If
is the function whose graph is
, find the values of
,
,
,
rough graph of .