2Integration
IA Differential Equations
2.1 Integration
Definition (Integral). An integral is the limit of a sum, e.g.
Z
b
a
f(x) dx = lim
∆x→0
N
X
n=0
f(x
n
)∆x.
For example, we can take ∆
x
=
b−a
N
and
x
n
=
a
+
n
∆
x
. Note that an integral
need not be defined with this particular ∆
x
and
x
n
. The term “integral” simply
refers to any limit of a sum (The usual integrals we use are a special kind known
as Riemann integral, which we will study formally in Analysis I). Pictorially, we
have
x
y
a x
1
x
2
x
3
···
x
n
x
n+1
···
···
b
The area under the graph from
x
n
to
x
n+1
is
f
(
x
n
)∆
x
+
O
(∆
x
2
). Provided
that f is differentiable, the total area under the graph from a to b is
lim
N→∞
N−1
X
n=0
(f(x
n
)∆x)+N·O(∆x
2
) = lim
N→∞
N−1
X
n=0
(f(x
n
)∆x)+O(∆x) =
Z
b
a
f(x) dx
Theorem (Fundamental Theorem of Calculus). Let
F
(
x
) =
R
x
a
f
(
t
) d
t
. Then
F
0
(x) = f(x).
Proof.
d
dx
F (x) = lim
h→0
1
h
"
Z
x+h
a
f(t) dt −
Z
x
a
f(t) dt
#
= lim
h→0
1
h
Z
x+h
x
f(t) dt
= lim
h→0
1
h
[f(x)h + O(h
2
)]
= f(x)
Similarly, we have
d
dx
Z
b
x
f(t) dt = −f(x)
and
d
dx
Z
g(x)
a
f(t) dt = f(g(x))g
0
(x).
Notation. We write
R
f
(
x
) d
x
=
R
x
f
(
t
) d
t
, where the unspecified lower limit
gives rise to the constant of integration.