# Integrating Log(x²+y²)

I'm making the following calculations:

X,Y,L=var('X,Y,L')

assume(L>0)

F(L)=integrate(integrate(log((X)^2+(Y)^2),Y,L/2,3*L/2),X,-L/2,L/2)

Maxima requested tor assume(4*X^2+L^2-4>0) and for assume(L-2>0), so I run the following

X,Y,L=var('X,Y,L')

assume(L>0)

assume(4*X^2+L^2-4>0)

assume(L-2>0)

F(L)=integrate(integrate(log((X)^2+(Y)^2),Y,L/2,3*L/2),X,-L/2,L/2)

print(F)

and the result is

-1/4piL^2 + 1/2L^2(arctan(3) - 9arctan(1/3) - 2) + 9L^2arctan(1/3) + 3/2L^2log(5/2L^2) - 1/2L^2log(1/2L^2) - 2L^2

After this, I change the limits of the integration

X,Y,L=var('X,Y,L')

assume(L>0)

assume(4*X^2+L^2-4>0)

assume(L-2>0)

F(L)=integrate(integrate(log((X)^2+(Y)^2),Y,-3*L/2,-L/2),X,-L/2,L/2)

print(F)

and the result is

1/4piL^2 - 1/2L^2(arctan(3) - 9arctan(1/3) - 2) - 9L^2arctan(1/3) - 3/2L^2log(5/2L^2) + 1/2L^2log(1/2L^2) + 2L^2

If we subtract this to results and considering L=10 The result should be Zero, but the result is very different

Z(L)=(-1/4piL^2 + 1/2L^2(arctan(3) - 9arctan(1/3) - 2) + 9L^2arctan(1/3) + 3/2L^2log(5/2L^2) - 1/2L^2log(1/2L^2) - 2L^2)-(1/4piL^2 - 1/2L^2(arctan(3) - 9arctan(1/3) - 2) - 9L^2arctan(1/3) - 3/2L^2log(5/2L^2) + 1/2L^2log(1/2L^2) + 2L^2)

Z(10).n()

922.636418333173

I know that the result should be zero from the math and I also use the software Mathematica.

The question is: I'm making something wrong? Or this is a well known problem of Sage?

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Since the arguments of the logarithm are squared, your second integral merely reverses the order of integration compared to the first. The second integral is the negative of the first: if you add them together you'll get zero.

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You are right, when we add the results we get zero. But this don't make sense. The subtraction should be zero, since L>0. So, this is something that I don't formulate very well in Sage? Or is something about my adaptation to Sage?

The answer depends only on L^2, so the sign of L won't change anything.