Example 8.   Let  [Graphics:Images/RiemannSumMod_gr_142.gif]  over  [Graphics:Images/RiemannSumMod_gr_143.gif].  
8 (a) Find the formula for the midpoint rule sum using n subintervals.
8 (b) Find the limit of the midpoint rule sum in part (a).

Solution 8 (a).

(a) Find the formula for the midpoint rule sum using n subintervals.

[Graphics:../Images/RiemannSumMod_gr_144.gif]


[Graphics:../Images/RiemannSumMod_gr_145.gif]
[Graphics:../Images/RiemannSumMod_gr_146.gif]
[Graphics:../Images/RiemannSumMod_gr_147.gif]

[Graphics:../Images/RiemannSumMod_gr_148.gif]

[Graphics:../Images/RiemannSumMod_gr_149.gif]

[Graphics:../Images/RiemannSumMod_gr_150.gif]

 

Solution 8 (b).

(b) Find the limit of the midpoint rule sum in part (a).

[Graphics:../Images/RiemannSumMod_gr_151.gif]

[Graphics:../Images/RiemannSumMod_gr_152.gif]

[Graphics:../Images/RiemannSumMod_gr_153.gif]
[Graphics:../Images/RiemannSumMod_gr_154.gif]

[Graphics:../Images/RiemannSumMod_gr_155.gif]

[Graphics:../Images/RiemannSumMod_gr_156.gif]

[Graphics:../Images/RiemannSumMod_gr_157.gif]

[Graphics:../Images/RiemannSumMod_gr_158.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004