How to compute this triple integral?

Question

In Mathematica how can I compute this integral:$$ \iiint_{D}\sqrt{(1-9z^2)(1-4y^2-9z^2)}\,dx\,dy\,dz$$ where D is the domain:

$$D: x^2 +4y^2+9z^2\le1$$

Please I need help!!!

Please, check the documentation of Integrate. Specifically, look into the Scope section, subsection Integrals over Regions — Sjoerd C. de Vries, Jun 03 '14 at 15:31
Can be this correct? Integrate[(Sqrt[(1 - 9 z^2) (1 - 9 z^2 - 4 y^2)]) Boole[-Sqrt[ 1 - x^2 - 4 y^2]/3 <= z <= Sqrt[1 - x^2 - 4 y^2]/3], {x, -1, 1}, {y, -1/2, 1/2}, {z, -1/3, 1/3}] — user14738, Jun 03 '14 at 15:47
Only for the sake of math: it you redefine variables as $X=x$, $Y=2y$ and $Z=3z$, you see that the domain is a sphere! — Mahdi, Jun 03 '14 at 17:08
Just use your domain exactly as you specified it in the question as an argument of Boole. Don't make it more complicated than necessary. — Sjoerd C. de Vries, Jun 03 '14 at 17:37

score 4 · Answer 1 · edited Apr 13 '17 at 12:55

4

This is perhaps too "creative". Some health checks needed for the series behavior:

Graphics`Region`RegionInit[];
region = (x x + 4 y y + 9 z z <= 1);
paregion = Region`ParametricRegion[{{x, y, z}, region}];
k = FullSimplify@
     Normal@Series[Sqrt[(1-9 z^2) (1-9 z^2-4 y^2)], {z,0, #},{y,0, #}] &/@ Range[1, 10, 2];
res = N@Integrate[#, {x, y, z} ∈ paregion] & /@ k

(* {0.698132, 0.488692, 0.480154, 0.477423, 0.476201} *)

So the result is near to 0.476

ListLinePlot@res

Mathematica graphics

This is where I've read first about this way for using Integrate[]

edited Apr 13 '17 at 12:55

Community

1

answered Jun 03 '14 at 16:57

Dr. belisarius

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I don't think this is the most practical way to integrate this. I'm posting it more as a curiosity. – Dr. belisarius Jun 03 '14 at 19:10
+1 for the undocumented integrate use :) – RunnyKine Jun 03 '14 at 19:10
@RunnyKine Thanks for remembering me that! Added a link. – Dr. belisarius Jun 03 '14 at 19:51

score 4 · Answer 2 · answered Jun 03 '14 at 18:04

4

I just did this:

Integrate[Boole[x^2 + 4*y^2 + 9*z^2 <= 1]*
 Sqrt[(1 - 9*z^2)*(1 - 4*y^2 - 9*z^2)], {z, -Infinity, Plus[Infinity]},{y, -Infinity, Plus[Infinity]}, {x, -Infinity, Plus[Infinity]}]

MMA quickly returned

64/135

answered Jun 03 '14 at 18:04

Cassini

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The order in which you did the integrations is important — doing x then y then z gives the quickest route through the nested integral. Other orderings seem not to evaluate. – Stephen Luttrell Jun 03 '14 at 19:42
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Heh. Sometimes it's better to be lucky than to be smart! – Cassini Jun 03 '14 at 19:56

How to compute this triple integral?

2 Answers2