Example 6.  Use Monte Carlo simulation to approximate the area of the cardioid defined by the constraint

        [Graphics:Images/MonteCarloPiMod_gr_159.gif].

Solution 6.

Set up the constraint for the cardioid.  

[Graphics:../Images/MonteCarloPiMod_gr_160.gif]


[Graphics:../Images/MonteCarloPiMod_gr_161.gif]

[Graphics:../Images/MonteCarloPiMod_gr_162.gif]

 

Set up the rectangular box enclosing the area.

 

[Graphics:../Images/MonteCarloPiMod_gr_163.gif]


[Graphics:../Images/MonteCarloPiMod_gr_164.gif]

[Graphics:../Images/MonteCarloPiMod_gr_165.gif]

Explore what happens with  [Graphics:../Images/MonteCarloPiMod_gr_166.gif]  points.

 

 

 

[Graphics:../Images/MonteCarloPiMod_gr_167.gif]

[Graphics:../Images/MonteCarloPiMod_gr_168.gif]

[Graphics:../Images/MonteCarloPiMod_gr_169.gif]

 

 

 

[Graphics:../Images/MonteCarloPiMod_gr_170.gif]

[Graphics:../Images/MonteCarloPiMod_gr_171.gif]

[Graphics:../Images/MonteCarloPiMod_gr_172.gif]

 

 

 

[Graphics:../Images/MonteCarloPiMod_gr_173.gif]

[Graphics:../Images/MonteCarloPiMod_gr_174.gif]

[Graphics:../Images/MonteCarloPiMod_gr_175.gif]

 

 

We are done.  

Aside.  The analytic value of the area can be found using polar coordinates.

[Graphics:../Images/MonteCarloPiMod_gr_176.gif]



[Graphics:../Images/MonteCarloPiMod_gr_177.gif]

[Graphics:../Images/MonteCarloPiMod_gr_178.gif]

[Graphics:../Images/MonteCarloPiMod_gr_179.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005