Example 1.  Numerically approximate the integral  [Graphics:Images/MidpointRuleMod_gr_47.gif]  by using the midpoint rule with  m = 1, 2, 4, 8, and 16  subintervals.

Solution 1.

[Graphics:../Images/MidpointRuleMod_gr_48.gif]

[Graphics:../Images/MidpointRuleMod_gr_49.gif]

[Graphics:../Images/MidpointRuleMod_gr_50.gif]

We will use simulated hand computations for the solution.

[Graphics:../Images/MidpointRuleMod_gr_51.gif]

[Graphics:../Images/MidpointRuleMod_gr_52.gif]
[Graphics:../Images/MidpointRuleMod_gr_53.gif]
[Graphics:../Images/MidpointRuleMod_gr_54.gif]


[Graphics:../Images/MidpointRuleMod_gr_55.gif]
[Graphics:../Images/MidpointRuleMod_gr_56.gif]
[Graphics:../Images/MidpointRuleMod_gr_57.gif]


[Graphics:../Images/MidpointRuleMod_gr_58.gif]
[Graphics:../Images/MidpointRuleMod_gr_59.gif]
[Graphics:../Images/MidpointRuleMod_gr_60.gif]


[Graphics:../Images/MidpointRuleMod_gr_61.gif]
[Graphics:../Images/MidpointRuleMod_gr_62.gif]
[Graphics:../Images/MidpointRuleMod_gr_63.gif]


[Graphics:../Images/MidpointRuleMod_gr_64.gif]
[Graphics:../Images/MidpointRuleMod_gr_65.gif]
[Graphics:../Images/MidpointRuleMod_gr_66.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004