Example 6.   Let  [Graphics:Images/RiemannSumMod_gr_96.gif]  over  [Graphics:Images/RiemannSumMod_gr_97.gif].  
6 (a) Find the formula for the left Riemann sum using n subintervals.
6 (b) Find the limit of the left Riemann sum in part (a).

 

Solution 6 (a).

(a) Find the formula for the left Riemann sum using n subintervals.

[Graphics:../Images/RiemannSumMod_gr_98.gif]

[Graphics:../Images/RiemannSumMod_gr_99.gif]
[Graphics:../Images/RiemannSumMod_gr_100.gif]
[Graphics:../Images/RiemannSumMod_gr_101.gif]

[Graphics:../Images/RiemannSumMod_gr_102.gif]

[Graphics:../Images/RiemannSumMod_gr_103.gif]

[Graphics:../Images/RiemannSumMod_gr_104.gif]
[Graphics:../Images/RiemannSumMod_gr_105.gif]
[Graphics:../Images/RiemannSumMod_gr_106.gif]
[Graphics:../Images/RiemannSumMod_gr_107.gif]
[Graphics:../Images/RiemannSumMod_gr_108.gif]

[Graphics:../Images/RiemannSumMod_gr_109.gif]

 

Solution 6 (b).
(b) Find the limit of the left Riemann sum in part (a).

[Graphics:../Images/RiemannSumMod_gr_110.gif]

[Graphics:../Images/RiemannSumMod_gr_111.gif]

[Graphics:../Images/RiemannSumMod_gr_112.gif]
[Graphics:../Images/RiemannSumMod_gr_113.gif]

[Graphics:../Images/RiemannSumMod_gr_114.gif]

[Graphics:../Images/RiemannSumMod_gr_115.gif]

[Graphics:../Images/RiemannSumMod_gr_116.gif]

[Graphics:../Images/RiemannSumMod_gr_117.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004