214 CHAPTER 9. INTEGRATION
Definition 9.9.3 When a function defined on an interval is either increasing on theinterval or decreasing on the interval, call the function monotone.
Theorem 9.9.4 Let f : [a,b]× [c,d]→ R be continuous and let β ,α be monotonefunctions on [c,d] , [a,b] respectively. Then
∫ b
a
∫ d
cf (x,y)dβ (y)dα (x) =
∫ d
c
∫ b
af (x,y)dα (x)dβ (y)
Proof: ∫ b
a
∫ d
cf (x,y)dβ (y)dα (x) =
n
∑i=1
∫ xi
xi−1
∫ d
cf (x,y)dβ (y)dα (x)
=n
∑i=1
∫ xi
xi−1
m
∑j=1
∫ y j
y j−1
f (x,y)dβ (y)dα (x) =n
∑i=1
m
∑j=1
∫ xi
xi−1
∫ y j
y j−1
f (x,y)dβ (y)dα (x)
By the mean value theorem for integrals, Theorem 9.9.1, this is
n
∑i=1
m
∑j=1
∫ xi
xi−1
(β (y j)−β
(y j−1
))f (x, t j)dα (x)
=n
∑i=1
m
∑j=1
(β (y j)−β
(y j−1
))(α (xi)−α (xi−1)) f (si, t j)
Also, by the same reasoning,
∫ d
c
∫ b
af (x,y)dα (x)dβ (y) =
m
∑j=1
n
∑i=1
(β (y j)−β
(y j−1
))(α (xi)−α (xi−1)) f
(s′i, t′j)
and now because of uniform continuity, it follows that if the partition points are closeenough, ∣∣ f (s′j, t ′j)− f (s j, t j)
∣∣< ε
(β (d)−β (c))(α (b)−α (a))
and so∣∣∣∫ d
c∫ b
a f (x,y)dα (x)dβ (y)−∫ b
a∫ d
c f (x,y)dβ (y)dα (x)∣∣∣ < ε . Since ε is arbitrary,
this shows the two iterated integrals are equal.The following is concerning a very important formula. First recall the arctan function.
Restricting tan to(−π
2 ,π
2
), this function is one to one and has an inverse function called
arctan. Thus arctan(y) = x where x ∈(−π
2 ,π
2
)and tanx = y. Then, using the theory of the
derivative of inverse functions, it follows that arctan is differentiable and arctan′ (y)y′ (x) =1 and so arctan′ (y) = 1
sec2(x) =1
1+tan2(x) =1
1+y2 . Also, arctan(0) = 0 obviously because
tan(0) = 0. Therefore, arctan(y) =∫ y
01
1+u2 du.Incidentally, this nice formula can be used to obtain all the trig functions. Note that
arctan(1) = π/4 because, from the above development, tan(π/4) = 1.
Theorem 9.9.5 The following holds. limx→∞
∫ x0 e−t2
dt =√
π
2 .