Example 2.  Find the analytic solution to the iterated integral  [Graphics:Images/SimpsonsRule2DMod_gr_67.gif].  

Solution 2.

Integrate  [Graphics:../Images/SimpsonsRule2DMod_gr_68.gif]  with respect to the variable y.  

[Graphics:../Images/SimpsonsRule2DMod_gr_69.gif]


[Graphics:../Images/SimpsonsRule2DMod_gr_70.gif]

Compute the definite integral  [Graphics:../Images/SimpsonsRule2DMod_gr_71.gif].

[Graphics:../Images/SimpsonsRule2DMod_gr_72.gif]


[Graphics:../Images/SimpsonsRule2DMod_gr_73.gif]

[Graphics:../Images/SimpsonsRule2DMod_gr_74.gif]

Integrate  [Graphics:../Images/SimpsonsRule2DMod_gr_75.gif]  with respect to the variable x.  

[Graphics:../Images/SimpsonsRule2DMod_gr_76.gif]


[Graphics:../Images/SimpsonsRule2DMod_gr_77.gif]

Compute the definite integral  [Graphics:../Images/SimpsonsRule2DMod_gr_78.gif]

[Graphics:../Images/SimpsonsRule2DMod_gr_79.gif]


[Graphics:../Images/SimpsonsRule2DMod_gr_80.gif]

[Graphics:../Images/SimpsonsRule2DMod_gr_81.gif]

We have found the value of the iterated integral

[Graphics:../Images/SimpsonsRule2DMod_gr_82.gif]


[Graphics:../Images/SimpsonsRule2DMod_gr_83.gif]

[Graphics:../Images/SimpsonsRule2DMod_gr_84.gif]

[Graphics:../Images/SimpsonsRule2DMod_gr_85.gif]

[Graphics:../Images/SimpsonsRule2DMod_gr_86.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004