So have this code (giving the moment of inertia of a cube)

Integrate[x^2 + y^2, 
          {x, y, z} ∈ ImplicitRegion[Max[Abs[x], Abs[y], Abs[z]] < 1, {x, y, z}]]

which works fine giving 16/3. But when I replace 1 by an abstract parameter a like in

Integrate[x^2 + y^2, 
          {x, y, z} ∈ ImplicitRegion[Max[Abs[x], Abs[y], Abs[z]] < a, {x, y, z}]]

I only get something obvious like the following: Mathematica output

How can I make Mathematica use the formal parameter?

  • $\begingroup$ If you would not need implicit regions you could do this: Integrate[ x^2 + y^2, {x, y, z} \[Element] Cuboid[a {-1, -1, -1}, a {1, 1, 1}]] which leads to: ConditionalExpression[(16 a^5)/3, a > 0] but that will probably not solve your problem $\endgroup$
    – Ruud3.1415
    Jul 21, 2017 at 13:48
  • $\begingroup$ I'm aware of Mathematica's predefined Regions, but I do need ImplicitRegion for my actual problem $\endgroup$
    – chr
    Jul 21, 2017 at 14:05
  • $\begingroup$ That's what I thought (since this problem is of course solvable without the regions). I thought putting the assumption in that a is larger than 0 would help solve it, but that also did not seem to work. $\endgroup$
    – Ruud3.1415
    Jul 21, 2017 at 14:10
  • $\begingroup$ Adding that assumption also was my first try $\endgroup$
    – chr
    Jul 21, 2017 at 15:21

2 Answers 2


maybe rewritting your implicit region as:

Integrate[x^2 + y^2, {x, y, z} \[Element] 
  ImplicitRegion[-a <= x <= a && -a <= y <= a && -a <= z <= a, {x, y, 
    z}], Assumptions -> a > 0]

(* (16 a^5)/3 *)
  • $\begingroup$ Thanks! Seems that Region doesn't like Abs $\endgroup$
    – chr
    Jul 31, 2017 at 17:17
  • 1
    $\begingroup$ Well, Region does like Abs :-) It needs a clear definition of the region: region = ImplicitRegion[Max[Abs[{x, y, z}]] < a, {{x, -a, a}, {y, -a, a}, {z, -a, a}}], then: Integrate[x^2 + y^2, Element[{x, y, z}, region], Assumptions -> a > 0] which gives ((16 a^5)/3). $\endgroup$
    – djg
    Jul 31, 2017 at 23:54

MMA tries to figure out the implicit region without no assumptions about $\{x,y,z\}$ or $a$, so we can tell it what to do in specific cases:

 Integrate[x^2 + y^2, {x, y, z} \[Element] ImplicitRegion[
 Evaluate@Reduce[Max[Abs[x], Abs[y], Abs[z]] < a, {x, y, z}, Reals], {x, y, z}]]

$\begin{array}{cc} \bigg\{ & \begin{array}{cc} \frac{16 a^5}{3} & a>0 \\ 0 & \text{True} \\ \end{array} \\ \end{array}$


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