Example 12.  Let  [Graphics:Images/MonteCarloMod_gr_264.gif].  Use the Monte Carlo method to calculate approximations to the integral  

        [Graphics:Images/MonteCarloMod_gr_265.gif].  

Solution 12.

[Graphics:../Images/MonteCarloMod_gr_266.gif]



[Graphics:../Images/MonteCarloMod_gr_267.gif]

 

 

 

[Graphics:../Images/MonteCarloMod_gr_268.gif]


[Graphics:../Images/MonteCarloMod_gr_269.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 MonteCarloIntegral4D.

[Graphics:../Images/MonteCarloMod_gr_270.gif]

[Graphics:../Images/MonteCarloMod_gr_271.gif]


[Graphics:../Images/MonteCarloMod_gr_272.gif]

[Graphics:../Images/MonteCarloMod_gr_273.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005