# A triple integral involving Abs over an ellipsoidal region

I'm a newbie and I'm trying to calculate a triple integral. But Mathematica doesn't output for half an hour and the CPU occupancy rate of my Wolfram doesn't changed when it's calculating. Here is the code:

f[x_, y_, z_] = Abs[x + y + z];
Integrate[f[x, y, z] Boole[x^2 + y^2 + 4 z^2 <= 1], {x, -1, 1}, {y, -1, 1},{z, -0.5, 0.5}]


I think it shouldn't spend a lot time to calculate this; I Googled and searched, but I still can't solve. Please help me. Thanks!

• Is a numerical value okay? Try NIntegrate rather than Integrate. It will give you a number in half a second. – march Oct 23 '15 at 3:50
• After a change of (ellipsoidal) coordinates, the integral becomes Integrate[Abs[Cos[v] + 2 (Cos[u] + Sin[u]) Sin[v]] Sin[v], {u, 0, 2 π}, {v, 0, π}]/16. At least in this case, Mathematica is faster in determining that it doesn't have a clue on how to deal with it symbolically. FWIW, the integral can be expressed using the new region functionality: Integrate[Abs[x + y + z], {x, y, z} ∈ Ellipsoid[{0, 0, 0}, {1, 1, 1/2}]]. – J. M. is in limbo Oct 23 '15 at 5:30
• You can transform the ellipsoid region to a unit ball, and align the norm of plane $x+y+z=0$ (should be something like $x+y+\zeta/2 = 0$ under new coord system) to the new $z$ axis, then you'll find yourself a very simple integral which can be symbolically done by Mathematica in seconds to get an exact answer $\frac{3 \pi }{8}$. – Silvia Oct 23 '15 at 6:42
• @Silvia How about giving an answer :) ? – xzczd Nov 10 '15 at 5:51
• Here's one way to do @Silvia's lovely approach: tr = Composition[ScalingTransform[{1, 1, 1/2}], RotationTransform[{{0, 0, 1}, {1, 1, 1/2}}]]; Integrate[(Abs[x + y + z] /. Thread[{x, y, z} -> tr[{x, y, z}]]) Det[D[tr[{x, y, z}], {{x, y, z}}]], {x, y, z} ∈ Ball[]] – J. M. is in limbo Nov 10 '15 at 7:30

RegionPlot3D[ • How does this answer the question about Integrate? – Daniel Lichtblau Aug 4 '17 at 12:55