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. , .

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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

      .