Calculus-Sec-5-2-Solution


5.2   The Definite Integral                 by SGLee - HSKim- SWSun

                                                                                                                 http://youtu.be/iUsf1h_hTAE

 

1-4. Find the Riemann sum by using the Midpoint Rule with the given value of  to approximate the integral.

 

1. 

http://matrix.skku.ac.kr/cal-lab/cal-RiemannSum.html

http://matrix.skku.ac.kr/Mobile-Sage-G/sage-grapher-riemann_sum.html

 

    Let . With  the interval width is  and midpoints are 

    for . So the Riemann sum is

    

 

                                        




2. 

 

 

      Let . With  the interval width is  and midpoints are

       for . So the Riemann sum is

      




3. 

 

     




 4. .

http://matrix.skku.ac.kr/cal-lab/cal-5-2-4.html

 

                                    




5-8. Express the limit as a definite integral on the given interval.

 

5. 

 

     .




6. 

 

     .

7. 

 

 

    .




8. 

 

     .




9-18. Determine whether the statement is true or false. If it is true, explain why. If it is false, give a counterexample.

 

9. If  and  are continuous on , then 

                                    .

 

 

     True by Definition

10. If  and  are continuous on , then

                                .

 

      False : A counterexample is   and .




11. If  and  are continuous on  and  for all , then

                                  .

 

 

      False : counterexample.   and .




12. If  is continuous on , then 

                                     .

 

 

      True




13. If  is continuous on , then

                                    .

 

      False




14. If  is continuous on  and , then 

                                .

 

      False

      Let . Then  and .

      Hence .




15. If  then  for all .

 

 

      False.  




16. If  and  are continuous and  and  then 

                                                .

 

 

      Let , and    (). Then

     . Because 

       , . So 

     Therefore .




17. If  and  are differentiable and  for , then  for .

 

 

      False

 18. All continuous functions are integrable.


 

      Yes

19-21. Evaluate the integral. (You should mention which method you are using.)

 

19. 

       http://matrix.skku.ac.kr/cal-lab/cal-5-2-19.html 


 

     = .




20. 

      http://matrix.skku.ac.kr/cal-lab/cal-5-2-20.html 

 

 

    




21. 

 

 

    .




22. 

 

 

    .




23-26. Evaluate the integral by interpreting as a sum of the areas.

 

23. 

 

 

    .

                                     




24. 

 

 

    .

                                      




25. 

 

 

      Let . Then .

      Since  and ,   we have .

                                 




26. 

 

 

      Let . Then .

      Since  and , we have .

                                  




27. Prove that 

     http://matrix.skku.ac.kr/cal-lab/cal-RiemannSum.html

 

 

      By using the endpoint rule,

      

                   .

      Hence, .

28. Prove that 

 

 

    By using the endpoint rule,

      

                    

                    

                    

      Hence, .

29.If  and , find .

 

  

    .

30. If  and , find .

 

 

    .

31. Find  if 

 

     Since  is continuous.

     

                                

32-35. Verify the inequality without evaluating the integrals.

 

32. 

 

 

      Since  for , we have .

      Hence, .

      http://matrix.skku.ac.kr/cal-lab/cal-5-2-32.html 




33. 

 

     Since  for , we obtain .

     Hence .




34. 

 

       Since  for , we obtain

      .

      Hence, .




35. 

 

 

     Since  and  for , we obtain

      .




                                   

                                           14th-century Korean Celestial Map (Cheonsang Yeolcha Bunyajido)

 

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