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. 
http://matrix.skku.ac.kr/cal-lab/cal-5-2-19.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, 
.
33. 
Since 
for 
, we obtain 
.
Hence 
.
34. 
Since 
for 
, we obtain
.
Hence, 
.
35. 
Since 
and 
for 
, we obtain
.
4. 
, 
.
