Example 2.  Let  [Graphics:Images/MonteCarloMod_gr_46.gif].  Use the midpoint rule to calculate approximations to the integral  [Graphics:Images/MonteCarloMod_gr_47.gif].

Solution 2.

[Graphics:../Images/MonteCarloMod_gr_48.gif]


[Graphics:../Images/MonteCarloMod_gr_49.gif]

[Graphics:../Images/MonteCarloMod_gr_50.gif]



[Graphics:../Images/MonteCarloMod_gr_51.gif]




[Graphics:../Images/MonteCarloMod_gr_52.gif]

[Graphics:../Images/MonteCarloMod_gr_53.gif]

[Graphics:../Images/MonteCarloMod_gr_54.gif]

 

 

 

[Graphics:../Images/MonteCarloMod_gr_55.gif]

[Graphics:../Images/MonteCarloMod_gr_56.gif]

[Graphics:../Images/MonteCarloMod_gr_57.gif]

 

 

 

[Graphics:../Images/MonteCarloMod_gr_58.gif]

[Graphics:../Images/MonteCarloMod_gr_59.gif]

[Graphics:../Images/MonteCarloMod_gr_60.gif]

 

 

 

[Graphics:../Images/MonteCarloMod_gr_61.gif]

[Graphics:../Images/MonteCarloMod_gr_62.gif]

[Graphics:../Images/MonteCarloMod_gr_63.gif]

 

 

We are done.  

Aside.  The analytic value of the integral can be found.

[Graphics:../Images/MonteCarloMod_gr_64.gif]


[Graphics:../Images/MonteCarloMod_gr_65.gif]

[Graphics:../Images/MonteCarloMod_gr_66.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005