Advanced book on Mathematics Olympiad

Real Analysis 571

D

Figure 71

J=−

1

(u^2 +v^2 )^2

.

Therefore,
∫∫

D

dxdy

(x^2 +y^2 )^2

=

∫∫

D 1

dudv=

1

12

.

(D. Flondor, N. Donciu,Algebra ̧ ̆si Analiz ̆a Matematica ̆(Algebra and Mathematical
Analysis), Editura Didactica ̧ ̆si Pedagogica, Bucharest, 1965) ̆

517.In the equation of the curve that bounds the domain

(

x^2

a^2

+

y^2

b^2

) 2

=

x^2

a^2

−

y^2

b^2

,

the expression on the left suggests the use of generalized polar coordinates, which are
suited for elliptical domains. And indeed, if we setx=arcosθandy=brsinθ, the
equation of the curve becomesr^4 =r^2 cos 2θ,orr=

√

cos 2θ. The conditionx ≥ 0
becomes−π 2 ≤θ≤ π 2 , and because cos 2θshould be positive we should further have
−π 4 ≤θ≤π 4. Hence the domain of integration is

{

(r, θ ); 0 ≤r≤

√

cos 2θ,−

π

4

≤θ≤

π

4

}

.

The Jacobian of the transformation isJ=abr. Applying the formula for the change of
variables, the integral becomes

∫ π 4

−π 4

∫√cos 2θ

0

a^2 b^2 r^3 cosθ|sinθ|drdθ=

a^2 b^2

4

∫ π 4

0

cos^22 θsin 2θdθ=

a^2 b^2

24

.