Example 6.  Numerically approximate the integral [Graphics:Images/MidpointRuleMod_gr_112.gif] by using the midpoint rule with  m = 1, 2, 4, 8, and 16  subintervals.

Solution 6.

[Graphics:../Images/MidpointRuleMod_gr_113.gif]

[Graphics:../Images/MidpointRuleMod_gr_114.gif]

[Graphics:../Images/MidpointRuleMod_gr_115.gif]

We will use simulated hand computations for the solution.

[Graphics:../Images/MidpointRuleMod_gr_116.gif]
[Graphics:../Images/MidpointRuleMod_gr_117.gif]
[Graphics:../Images/MidpointRuleMod_gr_118.gif]


[Graphics:../Images/MidpointRuleMod_gr_119.gif]
[Graphics:../Images/MidpointRuleMod_gr_120.gif]
[Graphics:../Images/MidpointRuleMod_gr_121.gif]


[Graphics:../Images/MidpointRuleMod_gr_122.gif]
[Graphics:../Images/MidpointRuleMod_gr_123.gif]
[Graphics:../Images/MidpointRuleMod_gr_124.gif]


[Graphics:../Images/MidpointRuleMod_gr_125.gif]
[Graphics:../Images/MidpointRuleMod_gr_126.gif]
[Graphics:../Images/MidpointRuleMod_gr_127.gif]


[Graphics:../Images/MidpointRuleMod_gr_128.gif]
[Graphics:../Images/MidpointRuleMod_gr_129.gif]
[Graphics:../Images/MidpointRuleMod_gr_130.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004