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If I had an integral specified like this: $$\int_{0}^{1}dx \int_{2}^{3} dy \int_{y^2+5}^{y-4} xyz \ dz$$

I would use this command in WolframAlpha to calculate it:

integrate (xyz) dz dy dx, x=0 to 1, y=2 to 3, z=y^2+5 to y-4

However, what if I have the domain of integration specified as e.g. inequality, for example:

$$\iiint_{V} xyz \ dx \ dy \ dz$$ where: $\sqrt{x^2+y^2} \le z \le \sqrt{1-x^2-y^2}$

How can I input this to WolframAlpha to calculate the integral, without manually determining the domain of integration for each of the variables? Will it require Mathematica syntax?

If this can't be done in WolframAlpha, what other software can do this?

Note: I'm not looking for an answer to this integral, I want to find a general method to calculate such types of integrals using software.

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  • $\begingroup$ take a look here and try the same code in WA. $\endgroup$ – Masacroso May 16 '18 at 13:22
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To my knowledge, it is not possible to use the Wolfram Language with Wolfram Alpha (to its full extent). However, you can use the free version of the Wolfram Programming Lab to evaluate valid Mathematica code. Your type of integral can be calculated in Mathematica with the following syntax

Integrate[
  x^2*y^2*z^2, 
  Element[{x, y, z}, 
    ImplicitRegion[Sqrt[x^2 + y^2] <= z <= Sqrt[1 - x^2 - y^2], {x, y, z}]
  ]
]

(* ((128 - 71 Sqrt[2]) \[Pi])/60480 *)

If Mathematica can find an analytical solution, however, depends of course how hard the problem is.

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