238 CHAPTER 10. IMPROPER INTEGRALS, STIRLING’S FORMULA

7. Suppose f is a real valued function defined on [a,b]× [c,d] which is uniformly con-tinuous as described in Problem 6 and bounded which follow from an assumptionthat it is continuous. Show

x →∫ d

cf (x,y)dy, y →

∫ b

af (x,y)dx

are both continuous functions. The idea is you fix one of the variables, x in thefirst and then integrate the continuous function of y obtaining a real number whichdepends on the value of x fixed. Explain why it makes sense to write∫ b

a

∫ d

cf (x,y)dydx,

∫ d

c

∫ b

af (x,y)dxdy.

Now consider the first of the above iterated integrals. (That is what these are called.)Consider the following argument in which you fill in the details.∫ b

a

∫ d

cf (x,y)dydx =

n

∑i=1

∫ xi

xi−1

∫ d

cf (x,y)dydx

=n

∑i=1

∫ xi

xi−1

m

∑j=1

∫ y j

y j−1

f (x,y)dydx =n

∑i=1

m

∑j=1

∫ xi

xi−1

∫ y j

y j−1

f (x,y)dydx

=n

∑i=1

m

∑j=1

∫ xi

xi−1

(y j − y j−1

)f (x, t j)dx

=n

∑i=1

m

∑j=1

(y j − y j−1

)(xi − xi−1) f (si, t j)

Also ∫ d

c

∫ b

af (x,y)dxdy =

m

∑j=1

n

∑i=1

(y j − y j−1

)(xi − 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

(d − c)(b−a)

and so ∣∣∣∣∫ d

c

∫ b

af (x,y)dxdy−

∫ b

a

∫ d

cf (x,y)dydx

∣∣∣∣< ε

Since ε is arbitrary, this shows the two iterated integrals are equal. This is a case ofFubini’s theorem.

8. This problem is in Apostol [2]. Explain why whenever f is continuous on [a,b]

limn→∞

b−an

n

∑k=1

f(

a+ k(

b−an

))=∫ b

af dx.

Apply this to f (x) = 11+x2 on the interval [0,1] to obtain the very interesting formula

π

4 = limn→∞ ∑nk=1

nn2+k2 .