# Mathematica integral wrong

I need to evaluate the following integral:

 Integrate[1/Sqrt[(x - x1)^2 + (y - y1)^2 + (z - z1)^2],
Element[{x, y, z}, Ball[{0, 0, 0}, R]] ,
Element[{x1, y1, z1}, Ball[{0, 0, 0}, R1]],
Assumptions -> R > 0 && R1 > 0 ]


and Mathematica says the integral is 0, which is obviusly wrong. Am I doing something wrong?

• Wait, what does this mean? I want a double integral in $x,y,z$ and $x1, y1, z1$, with the former variables in a ball of radius $R$ and the latter in a ball of radius $R1$. In your suggestions, $R1$ doesn't show up Commented Jul 7 at 12:03
• I think his is looking for the gravitational energy between two balls, one placed inside the other—thus the double integral. @tommy1996q, I think there will be a “ball” of singularities in that intégrantd. But, I think your zero is probably not correct. (This comment referred to an earlier but deleted comment—i think the OP’s comment above also referred to the deleted comment) Commented Jul 7 at 12:04
• @CraigCarter nope, it's an entirely different problem, but it doesn't really matter. The point is that it's a very simple integration (conceptually at least) and I do not understand how Mathematica could give an answer so wrong. Trivially, I am integrating something which is always greater than 0. Commented Jul 7 at 12:07
• Greater than zero, yes, but infinite at points in the smaller ball—i think. The 0 seems like a bug in any case. Commented Jul 7 at 12:09
• @tommy1996q "The point is that it's a very simple integration..." So what is the correct answer when you know it is very simple? Commented Jul 7 at 16:15

Not an answer, but this won't fit in comment:

    Integrate[1/Sqrt[(x - x1)^2 + (y - y1)^2 + (z - z1)^2],
Element[{x, y, z},
RegionDifference[Ball[{0, 0, 0}, bigR], Ball[{0, 0, 0}, littleR]]],
Element[{x1, y1, z1}, Ball[{0, 0, 0}, littleR]],
Assumptions -> 0 < littleR < bigR]  (*gives 0*)


and

Integrate[1/Sqrt[(x - x1)^2 + (y - y1)^2 + (z - z1)^2],
Element[{x, y, z},
RegionDifference[Ball[{0, 0, 0}, bigR],
Ball[{0, 0, 0}, littleR - epsilon]]],
Element[{x1, y1, z1}, Ball[{0, 0, 0}, littleR]],
Assumptions -> 0 < littleR < bigR && 0 < epsilon < littleR] (*gives 0*)


and

NIntegrate[1/Sqrt[(x - x1)^2 + (y - y1)^2 + (z - z1)^2],
Element[{x, y, z},
RegionDifference[Ball[{0, 0, 0}, 2], Ball[{0, 0, 0}, 1 - 0.1]]],
Element[{x1, y1, z1}, Ball[{0, 0, 0}, 1 - 0.1]]
] (*gives 61.2 but with warnings about accuracy*)


I think that 0 is an incorrect result and worth a bug report.

NIntegrate[1/Sqrt[(x - x1)^2 + (y - y1)^2 + (z - z1)^2],
Element[{x, y, z}, Ball[{0, 0, 0}, 1]],
Element[{x1, y1, z1}, Ball[{0, 0, 0}, 2]]] (*gives 71.6 with warnings*)


but I don't trust that result at all.

• Thanks! Later I'll look more into it. Unluckily I'd really need some analytic solution. But yeah, 0 is most defnitily wrong. I'll report the bug Commented Jul 7 at 15:46