Calculus-Sec-5-2-Solution

5.2   The Definite Integral                 by SGLee - HSKim- SWSun

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.

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.

= .

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

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, .

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)