Advanced book on Mathematics Olympiad

176 3 Real Analysis

V=

1

2

∫∞

0

t^2

t^4 + 1

dt.

A routine but lengthy computation using Jacobi’s method of partial fraction decomposition
shows that the antiderivative of t
2
t^4 + 1 is
1
2

√

2

arctan

x^2 − 1

x

√

2

+

1

4

√

2

ln

x^2 −x

√

2 + 1

x^2 +x

√

2 + 1

+C,

whenceV = π

√

2

8. Equating the two values forV, we obtainI =

√

2 π

4. A similar

argument yieldsJ=

√ 2 π

4. 

The solutions to all but last problems below are based on appropriate changes of

coordinates.

514.Compute the integral

∫∫

Dxdxdy, where

D=

{

(x, y)∈R^2 |x≥ 0 , 1 ≤xy≤ 2 , 1 ≤

y

x

≤ 2

}

.

515.Find the integral of the function

f (x, y, z)=

x^4 + 2 y^4

x^4 + 4 y^4 +z^4

over the unit ballB={(x,y,z)|x^2 +y^2 +z^2 ≤ 1 }.

516.Compute the integral

∫∫

D

dxdy

(x^2 +y^2 )^2

,

whereDis the domain bounded by the circles

x^2 +y^2 − 2 x= 0 ,x^2 +y^2 − 4 x= 0 ,

x^2 +y^2 − 2 y= 0 ,x^2 +y^2 − 6 y= 0.

517.Compute the integral

I=

∫∫

D

|xy|dxdy,

where

D=

{

(x, y)∈R^2 |x≥ 0 ,

(

x^2

a^2

+

y^2

b^2

) 2

≤

x^2

a^2

−

y^2

b^2

}

,a,b> 0.