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Xnumbers Tutorial
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where
θ
, from 0 to 2
π
, is the rotation angle on the xy-plane
The triple integral changes in
∫∫∫
∫ ∫ ∫
=
D
z
z
f x x y y z z d d d d dz
dxdydz
f x x y y z
max
min
max
min
max
min
( , , , , )
( , , , , )
ρ
ρ
θ
θ
ρ ρ ρ θ
Functions f(x, y, z) and – eventually – also the boundary functions may be written in symbolic
expression..
For further details about the math string see Math formula string
Integration function can be:
• three-variate functions like x^2+y^2-x*z , log(1+x+y+z) , (x+z)/(1+x^2+y^2) , etc.
• Constant numbers like 0 , 2 , 1.5 , 1E-6, etc.
• Constant expressions like 1/2 , √2+1 , sin(pi/4) , etc.
Boundary limits can be:
• Constant numbers like 0 , 2 , 10 , 3.141 , etc.
• Constant expressions like 1/2 , √2+1, pi, sin(1/2*pi) , exp(1) , etc.
• univariate or bivariate functions like x/2 , 3y-10 , x^2+x-1 , sin(x+z), Ln(x+y) , etc.
A normal domain has, at least, two constant boundary limits.
The Integration function and the boundary limits can be passed to the macro directly or by
reference. That is: we can write directly the symbolic expressions into the input fields, or you
can pass the cells that containing the expressions (simpler), or even a mixed mode.
Let’s see how it works
Example 1. Approximate numerically the following triple integral
dxdydz
xyz
54
1
0
2
0
3
0
∫∫∫
The integration domain is the parallelepiped of lengths 1, 2, 3
The macro assumes as default the following simple layout (but, of course, it is not obligatory)
Now select the cell A2 containing the integration function and start the macro from the
Xnumbers toobar Macros > Integral > Triple.
