Example 7.  Use Monte Carlo simulation to approximate the area of the region defined by the constraints

        [Graphics:Images/MonteCarloPiMod_gr_180.gif]  

Solution 7.

Set up the constraints for the region.  

[Graphics:../Images/MonteCarloPiMod_gr_181.gif]


[Graphics:../Images/MonteCarloPiMod_gr_182.gif]

[Graphics:../Images/MonteCarloPiMod_gr_183.gif]

 

 

Set up the rectangular box enclosing the area.

[Graphics:../Images/MonteCarloPiMod_gr_184.gif]


[Graphics:../Images/MonteCarloPiMod_gr_185.gif]

[Graphics:../Images/MonteCarloPiMod_gr_186.gif]

Use Monte Carlo simulation  and explore what happens with  [Graphics:../Images/MonteCarloPiMod_gr_187.gif]  points.

 

 

 

[Graphics:../Images/MonteCarloPiMod_gr_188.gif]

[Graphics:../Images/MonteCarloPiMod_gr_189.gif]

[Graphics:../Images/MonteCarloPiMod_gr_190.gif]

 

 

 

[Graphics:../Images/MonteCarloPiMod_gr_191.gif]

[Graphics:../Images/MonteCarloPiMod_gr_192.gif]

[Graphics:../Images/MonteCarloPiMod_gr_193.gif]

 

 

 

[Graphics:../Images/MonteCarloPiMod_gr_194.gif]

[Graphics:../Images/MonteCarloPiMod_gr_195.gif]

[Graphics:../Images/MonteCarloPiMod_gr_196.gif]

 

 

We are done.  

Aside.  The analytic value of the area can be found by splitting it up into three parts.

The curved portion at the top from [Graphics:../Images/MonteCarloPiMod_gr_197.gif]  and the curved portion at the bottom from [Graphics:../Images/MonteCarloPiMod_gr_198.gif]  are given by the following formulas.

[Graphics:../Images/MonteCarloPiMod_gr_199.gif]



[Graphics:../Images/MonteCarloPiMod_gr_200.gif]

[Graphics:../Images/MonteCarloPiMod_gr_201.gif]

 

 

The curved portion at the left from [Graphics:../Images/MonteCarloPiMod_gr_202.gif]  and the curved portion at the right from [Graphics:../Images/MonteCarloPiMod_gr_203.gif]  are given by the following formulas.

[Graphics:../Images/MonteCarloPiMod_gr_204.gif]


[Graphics:../Images/MonteCarloPiMod_gr_205.gif]

[Graphics:../Images/MonteCarloPiMod_gr_206.gif]

 

 

 

[Graphics:../Images/MonteCarloPiMod_gr_207.gif]


[Graphics:../Images/MonteCarloPiMod_gr_208.gif]

[Graphics:../Images/MonteCarloPiMod_gr_209.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005