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 .

214 CHAPTER 9. INTEGRATIONDefinition 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] x {c,d] 3 R_ be continuous and let B,0 be monotonefunctions on |c,d] ,|a,b] respectively. Then[ [i rey280) do (x = [PP roaoe ) dB (y)Proof:i=[ [rena oraac=¥ [" ["roxaBoaawaXj yjyf yf foorab oda = PY [" [ fer)aB 0) dala1 j=1/j-1 i=] jal 7 Xi-1 YY j-1Also, by the same reasoning,[ [reesyaats dB (y = EY Br B (9;-1)) (oe (xi) — 0 (xi-1)) f (shat)and now because of uniform continuity, it follows that if the partition points are closeenough,E(B (a) —B (c)) (@(b) — @(a))and so i S? f (x,y) da (x) dB (y) — f? [4 f (x,y) dB (y)dae(x)| < €. Since é 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 (-3 5) z), this function i is one to one and has an inverse function calledarctan. Thus arctan (y) = x where x € (-% 55 5) and tanx = y. Then, using the theory of thederivative of inverse functions, it follows that arctan is differentiable and arctan’ (y) y’ (x) =1 and so arctan’ (y) = x2) = Tram = Tp Also, arctan (0) = 0 obviously becausetan (0) = 0. Therefore, arctan (y) = {i paadu.Incidentally, this nice formula can be used to obtain all the trig functions. Note thatarctan (1) = 2/4 because, from the above development, tan (2/4) = 1.If (84,04) —f(s;t,)| <Theorem 9.9.5 The following holds. lim, +.» Io et dt = VE