Example 8.  Let  [Graphics:Images/MonteCarloMod_gr_210.gif].  Use the Monte Carlo method to calculate approximations to the triple integral  

        [Graphics:Images/MonteCarloMod_gr_211.gif].  

Solution 8.

[Graphics:../Images/MonteCarloMod_gr_212.gif]



[Graphics:../Images/MonteCarloMod_gr_213.gif]

 

 

 

[Graphics:../Images/MonteCarloMod_gr_214.gif]


[Graphics:../Images/MonteCarloMod_gr_215.gif]

 

 

We are done!

Aside.  We can use Mathematica's subroutine NIntegrate to perform Monte Carlo integration, and we can verify that it is computing the same results as our subroutine MonteCarloIntegral3D.

[Graphics:../Images/MonteCarloMod_gr_216.gif]

[Graphics:../Images/MonteCarloMod_gr_217.gif]


[Graphics:../Images/MonteCarloMod_gr_218.gif]

[Graphics:../Images/MonteCarloMod_gr_219.gif]

Observe that the results are the same.

We are really done!  

Aside.  The analytic value of the integral can be found and involves the following steps.

[Graphics:../Images/MonteCarloMod_gr_220.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005