5.1   Sequences and Series                by SGLee - HSKim- SWSun


1. Find the area under the curve  from 0 to 2.


     We divide the interval [0,2] into n equal parts.

     Thus the length of each sub-interval is 2/n and the th sub-interval is given by 

      Now we apply the right end formula to find required area.



 2. Find the area of the region under the graph of  from 0 to 2.




3. Find the area under the curve  from  to , where .



      . (Since , the value what we evaluate is equal to the area.)

4. (a) Let  be the area of a polygon with  equal sides inscribed in a circle with radius .

      By dividing the polygon into  congruent triangles with central angle ,

      show that .

    (b) Show that  .

      [Hint: Use Equation 3.4.2]


   (a) The area of one piece of  

       consists of  such pieces so the area of 

      Remark: To estimate area under the graph of , one can take the sample point  as the point  such that .

      Similarly we can choose the sample point  as the point  such that .

   (b) Using ,


       (Let . Then  implies .) Think of the lower and upper sums of the graph.


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