Calculus-Sec-5-1-Solution


  5.1   Sequences and Series                by SGLee - HSKim- SWSun

                                                                                                            http://youtu.be/Y_nCn76RPmY 

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

     http://matrix.skku.ac.kr/cal-lab/cal-5-1-exs-1.html

 

     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.

            http://matrix.skku.ac.kr/cal-lab/cal-5-1-exs-2.html

 

     .







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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