Advanced book on Mathematics Olympiad

3.3 Multivariable Differential and Integral Calculus 177

518.Prove the Gaussian integral formula

∫∞

−∞

e−x

2

dx=

√

π.

519.Evaluate

∫ 1

0

∫ 1

0

∫ 1

0

( 1 +u^2 +v^2 +w^2 )−^2 dudvdw.

520.LetD={(x, y)∈R^2 | 0 ≤x≤y≤π}. Prove that

∫∫

D

ln|sin(x−y)|dxdy=−

π^2

2

ln 2.

Our next topic is the continuous analogue of the change of the order of summation

in a double sum.

Fubini’s theorem.Letf:R^2 →Rbe a piecewise continuous function such that

∫d

c

∫b

a

|f (x, y)|dxdy <∞.

Then

∫d

c

∫b

a

f (x, y)dxdy=

∫b

a

∫d

c

f (x, y)dydx.

The matter of convergence can be bypassed for positive functions, in which case we

have the following result.

Tonelli’s theorem.Letf:R^2 →Rbe a positive piecewise continuous function. Then

∫b

a

∫d

c

f (x, y)dxdy=

∫d

c

∫b

a

f (x, y)dydx.

The limits of integration can be finite or infinite. In the particular case thatf (x, y)is

constant on the squares of an integer lattice, we recover the discrete version of Fubini’s

theorem, the change of order of summation in a double sum

∑∞

m= 0

∑∞

n= 0

f (m, n)=

∑∞

n= 0

∑∞

m= 0

f (m, n).

A slightly more general situation occurs whenfis a step function in one of the variables.
In this case we recover the formula for commuting the sum with the integral: