Double integral

  • #1
587
10
If function is ##f(-x,-y)=f(x,y)##, is then

##\int^{a}_{-a}\int^{a}_{-a}f(x,y)dxdy=0##?
Thanks for answer.
 

Answers and Replies

  • #2
CompuChip
Science Advisor
Homework Helper
4,302
47
No, consider for example
$$\int_{-a}^a \int_{-a}^a x^2 y^2 \, dx dy$$
 
  • #3
587
10
Could you tell me some explanation why this is valid only for one integral?
 
  • #4
jgens
Gold Member
1,581
50
Could you tell me some explanation why this is valid only for one integral?
The result does not hold for 1-dimensional integrals either. If f(x) = x2, then f(x) = f(-x) and integrating over any interval of the form [-a,a] (where a ≠ 0) gives you a non-zero number.

If you have a function that satisfies f(-x,y) = -f(x,y) then the integral over [-a,a] × [-a,a] should be zero. So you should probably look for a condition like this.
 
  • #5
587
10
If function is ##f(-x,-y)=f(x,y)##, is then

##\int^{a}_{-a}\int^{a}_{-a}f(x,y)dxdy=0##?
Thanks for answer.

Sorry I thought about ##f(-x,-y)=-f(x,y)##

Is it always valid?
Also is it valid in case ##f(-x,-y)=f(x,y)## that
##\int^{a}_{-a}\int^{a}_{-a}f(x,y)dxdy=2\int^{a}_{0}f(x,y)dxdy##
 
  • #6
lurflurf
Homework Helper
2,432
132
For double integrals there are four cases
f(x,y)
f(-x,y)
f(x,-y)
f(-x,-y)

f(-x,-y)=-f(x,y)
implies that
f(x,-y)=-f(x,-y)
thus
the integral would be zero

f(-x,-y)=f(x,y)
implies
f(-x,y)=f(x,-y)
##\int^{a}_{-a}\int^{a}_{-a}f(x,y)dxdy=2\int^{a}_{-a}\int^{a}_{0}f(x,y)dxdy##
 
  • #7
587
10
Tnx.
 

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