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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!

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  • $\begingroup$ Is a numerical value okay? Try NIntegrate rather than Integrate. It will give you a number in half a second. $\endgroup$
    – march
    Commented Oct 23, 2015 at 3:50
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    $\begingroup$ 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}]]. $\endgroup$
    – J. M.'s missing motivation ♦
    Commented Oct 23, 2015 at 5:30
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    $\begingroup$ 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 $\frac{3 \pi }{8}$. $\endgroup$
    – Silvia
    Commented Oct 23, 2015 at 6:42
  • $\begingroup$ @Silvia How about giving an answer :) ? $\endgroup$
    – xzczd ♦
    Commented Nov 10, 2015 at 5:51
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    $\begingroup$ 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[]] $\endgroup$
    – J. M.'s missing motivation ♦
    Commented Nov 10, 2015 at 7:30

1 Answer 1

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This could be an option:

RegionPlot3D[
 x^2 + y^2 + 4 z^2 <= 1, {x, -1, 1}, {y, -1, 1}, {z, -0.5, 0.5}]

enter image description here

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    $\begingroup$ How does this answer the question about Integrate? $\endgroup$
    – Daniel Lichtblau
    Commented Aug 4, 2017 at 12:55

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