5.2 The Definite Integral


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

1. , .

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

1.04754618408

 



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


10. If and are continuous on , then

.

 False


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

 False


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 .

 True


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

 False


18. All continuous functions have derivatives.

 False


19-21. Evaluate the integral.


19. .  

 .


20. .

 .


21.

 .


22. .

 .


23-26. Evaluate the integral by interpreting in terms of the areas.


23. .

 .


24. .

 .


25. .

 Let . Then .

      Since and , we have .


26. .

 Let . Then .

      Since and , we have .

 


27. Prove that .

 By using the end point rule,


      

              .

      Hence, .


28. Prove that .

 By using the end point 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, .


33. .

 Since for , we obtain .

      Hence .


34. .

 Since for , we obtain

      .

      Hence, .


35. .

 Since and for , we obtain

      .