# Conditional numerical integration boundaries

I have a multidimensional integration of the form:

  somefunc[t] = NIntegrate[ otherfunc[x, y, z, t ],
{z, z1[t], z2[t]}, {y, y1[t, z], y2[t, z]}, {x, x1[t, z, y], x2[t, z, y]}];


This needs to be evaluated only when $z2 > z1$, $y2 > y1$ and $x2 > x1$. However mathematica still evaluates the integrand backwards even if $y2 < y1$. My question is how to tell NIntegrate to simply give 0 in such cases and move on the next integration step. I tried to modify the upper integration boundary as follows:

 y2p[t, z] = Max[y1[t, z], y2[t, z]]


But this is not accepted by Mathematica as an integration boundary. How can I fix this?

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Please post your complete and valid code. –  Mr.Wizard Mar 6 '12 at 10:10
Are you missing a pattern from the LHS? Looks like somefunc[t] = should be somefunc[t_] :=. It also looks like most answers have copied this problem without noticing it. –  Szabolcs Mar 13 '12 at 5:21

A better option than using Boole would be to use Piecewise. Using that you can define a function that returns 0 when your conditions aren't met and otherfunc otherwise.

So, define a function otherfunc2 and integrate that:

otherfunc2[x_, y_, z_, t_] :=
Piecewise[
{
{otherfunc[x, y, z, t],
z1[t] <=z<= z2[t] && y1[t, z] <=y<= y2[t, z] && x1[t, z, y] <=x<= x2[t, z, y]},
{0, True}
}
]

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Maybe you can try

   somefunc[t] = NIntegrate[
Boole[z1[t] <= z <= z2[t]
&& y1[t, z] <= y <= y2[t, z]
&& x1[t, z, y] <= x <= x2[t, z, y]]
otherfunc[x, y, z, t ],
{z, z0, z1},{y, y0, y1}, {x, x0, x1}];


where the integration limits are appropriate constants.

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The integral is well defined whether y1 > y2, y1 < y2, or one or both of y1 and y2 is complex (so that < and > have no meaning). That's why NIntegrate proceeds with integration in any of these cases.

Therefore if you want these cases to return a special value, such as 0, you should do it manually with something like this:

somefunc[t] := If[z1[t] < z2[t] && ..., NIntegrate[...], 0];


(Definitely use := instead of =.)

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Can't you do something like

somefunc[t] = NIntegrate[ otherfunc[x, y, z, t ],
{z, z1[t], Max[z1[t], z2[t]]}, {y, y1[t, z], Max[y1[t, z], y2[t, z]]},
{x, x1[t, z, y], Max[x1[t, z, y], x2[t, z, y]]}];

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