1.2 Integrals with Parameters
5
namely the order of operations of the limit and integral can be exchanged. This
property is called the finding limit under the integral sign or taking limit under
the integral sign.
Proof For any given y ∈ [c, d], to take y, and to make y + y ∈ [c, d], there is
ϕ(y + y) − ϕ(y) =
b
a
[ f (x, y + y) − f (x, y)]dx
|ϕ(y + y) − ϕ(y)| =
b
a
[ f (x, y + y) − f (x, y)]dx
≤
b
a
| f (x, y + y) − f (x, y)|dx
Since the function f (x, y) is continuous in the closed domain [a, b; c, d], so it is
bound to be uniformly continuous in this domain, that is to say, for any given ε > 0,
there must be δ > 0, such that for two arbitrary points (x 1 , y 1 ) and (x 2 , y 2 ) in D,
as long as |x 2 − x 1 | < δ, |y 2 − y 1 | < δ, the inequality | f (x 2 , y 2 ) − f (x 1 , y 1 )| < ε
holds.
If x 1 = x 2 = x, for given two points (x, y) and (x, y + y) in D, when
|(y + y) − y| = |y| < δ, then the inequality | f (x, y + y) − f (x, y)| < ε
holds too. Thereupon, for any y ∈ [c, d], to take y + y ∈ [c, d], and |y| < δ,
then there is
|ϕ(y + y) − ϕ(y)| ≤
b
a
| f (x, y + y) − f (x, y)|dx <
b
a
εdx = ε(b − a)
namely the function ϕ(y) is continuous in the closed interval [c, d].
Let y 0 ∈ [c, d], according to the definition of continuous function, there is
lim
y→y 0
b
a
f (x, y)dx = lim
y→y 0
ϕ(y) = ϕ(y 0 ) =
b
a
f (x, y 0 )dx =
b
a
lim
y→y 0
f (x, y)dx
This shows that the function f (x, y) satisfies the condition supposed by
Theorem 1.2.1, the operation between the integral and limit can be exchanged order.
Quod erat demonstrandum.
Theorem 1.2.2 (The interchangeability of integral order) If f (x, y) is continuous
in a rectangular domain D[a, b; c, d], then there is
d
c
dy
b
a
f (x, y)dx =
b
a
dx
d
c
f (x, y)dy
(1.2.3)
namely the integral order can be exchanged, this property is called the integrating
under the integral sign.
Proof Because the two iterated integrals on both sides of Eq. (1.2.3) are all equal to
the double integral
˜
D f (x, y)dxdy, Eq. (1.2.3) holds. Quod erat demonstrandum.
5
namely the order of operations of the limit and integral can be exchanged. This
property is called the finding limit under the integral sign or taking limit under
the integral sign.
Proof For any given y ∈ [c, d], to take y, and to make y + y ∈ [c, d], there is
ϕ(y + y) − ϕ(y) =
b
a
[ f (x, y + y) − f (x, y)]dx
|ϕ(y + y) − ϕ(y)| =
b
a
[ f (x, y + y) − f (x, y)]dx
≤
b
a
| f (x, y + y) − f (x, y)|dx
Since the function f (x, y) is continuous in the closed domain [a, b; c, d], so it is
bound to be uniformly continuous in this domain, that is to say, for any given ε > 0,
there must be δ > 0, such that for two arbitrary points (x 1 , y 1 ) and (x 2 , y 2 ) in D,
as long as |x 2 − x 1 | < δ, |y 2 − y 1 | < δ, the inequality | f (x 2 , y 2 ) − f (x 1 , y 1 )| < ε
holds.
If x 1 = x 2 = x, for given two points (x, y) and (x, y + y) in D, when
|(y + y) − y| = |y| < δ, then the inequality | f (x, y + y) − f (x, y)| < ε
holds too. Thereupon, for any y ∈ [c, d], to take y + y ∈ [c, d], and |y| < δ,
then there is
|ϕ(y + y) − ϕ(y)| ≤
b
a
| f (x, y + y) − f (x, y)|dx <
b
a
εdx = ε(b − a)
namely the function ϕ(y) is continuous in the closed interval [c, d].
Let y 0 ∈ [c, d], according to the definition of continuous function, there is
lim
y→y 0
b
a
f (x, y)dx = lim
y→y 0
ϕ(y) = ϕ(y 0 ) =
b
a
f (x, y 0 )dx =
b
a
lim
y→y 0
f (x, y)dx
This shows that the function f (x, y) satisfies the condition supposed by
Theorem 1.2.1, the operation between the integral and limit can be exchanged order.
Quod erat demonstrandum.
Theorem 1.2.2 (The interchangeability of integral order) If f (x, y) is continuous
in a rectangular domain D[a, b; c, d], then there is
d
c
dy
b
a
f (x, y)dx =
b
a
dx
d
c
f (x, y)dy
(1.2.3)
namely the integral order can be exchanged, this property is called the integrating
under the integral sign.
Proof Because the two iterated integrals on both sides of Eq. (1.2.3) are all equal to
the double integral
˜
D f (x, y)dxdy, Eq. (1.2.3) holds. Quod erat demonstrandum.