Example 9.   Let  [Graphics:Images/RiemannSumMod_gr_159.gif]  over  [Graphics:Images/RiemannSumMod_gr_160.gif].  
9 (a) Find the formula for the trapezoidal rule sum using n subintervals.
9 (b) Find the limit of the trapezoidal rule sum in part (a).

Solution 9 (a).

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

[Graphics:../Images/RiemannSumMod_gr_161.gif]


[Graphics:../Images/RiemannSumMod_gr_162.gif]
[Graphics:../Images/RiemannSumMod_gr_163.gif]
[Graphics:../Images/RiemannSumMod_gr_164.gif]

[Graphics:../Images/RiemannSumMod_gr_165.gif]

[Graphics:../Images/RiemannSumMod_gr_166.gif]

[Graphics:../Images/RiemannSumMod_gr_167.gif]

[Graphics:../Images/RiemannSumMod_gr_168.gif]

 

Solution 9 (b).

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

[Graphics:../Images/RiemannSumMod_gr_169.gif]

[Graphics:../Images/RiemannSumMod_gr_170.gif]

[Graphics:../Images/RiemannSumMod_gr_171.gif]
[Graphics:../Images/RiemannSumMod_gr_172.gif]

[Graphics:../Images/RiemannSumMod_gr_173.gif]

[Graphics:../Images/RiemannSumMod_gr_174.gif]

[Graphics:../Images/RiemannSumMod_gr_175.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004