Example 14.  Let  [Graphics:Images/MonteCarloMod_gr_284.gif].  Use the Monte Carlo method to calculate approximations to the integral  

        [Graphics:Images/MonteCarloMod_gr_285.gif].  
        
Compare the Monte Carlo method with the midpoint rule.

Solution 14.

[Graphics:../Images/MonteCarloMod_gr_286.gif]



[Graphics:../Images/MonteCarloMod_gr_287.gif]

 

 

[Graphics:../Images/MonteCarloMod_gr_288.gif]


[Graphics:../Images/MonteCarloMod_gr_289.gif]

 

 

 

[Graphics:../Images/MonteCarloMod_gr_290.gif]


[Graphics:../Images/MonteCarloMod_gr_291.gif]

 

Remark.  For this quintuple multiple integral,  most of the time the Monte Carlo simulation has less error than the midpoint rule.  This is what we expect to happen.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005