MTH 252 Midterm Review 1. Estimate the area under the graph of ! ! = cos ! from ! = 0 to ! =
approximating rectangles and right endpoints.
!
!
using four
2. Estimate the area under the graph of ! ! = ! from ! = 1 to ! = 4 using six approximating
rectangles and left endpoints.
3. Estimate the area under the graph of ! ! = 1 + ! ! from ! = −1 to ! = 2 using three
approximating rectangles and midpoints.
4. Evaluate the Riemann sum for ! ! =
using right endpoints.
! !
!
on the interval −4, 0 with four subintervals and
5. Evaluate the Riemann sum for ! ! = ! ! − 2! on the interval −1, 3 with five subintervals and
using left endpoints.
6. Evaluate the Riemann sum for ! ! = 1 − ! on the interval 2,10 , with four subintervals and
using midpoints.
7. The speed of a runner (in feet per second) increased steadily during the first three seconds of a
race. Her speed at half-second intervals is given in the table. Approximate the distance the
runner traveled in the first three seconds of the race.
!
!(!)
0
0
0.5
6.2
1.0
10.8
1.5
14.9
2.0
18.1
2.5
19.4
3.0
20.2
8. Oil leaked from a tank at a rate of !(!) liters per hour. The rate decreased as time passed and
values of the rate at two-hour time intervals are shown in the table. Approximate the total
amount of oil that leaked from the tank.
!
!(!)
0
8.7
2
7.6
9. Estimate the area under the curve
from ! = 0 to ! = 7 using fourteen
subintervals and midpoints. 4
6.8
6
6.2
8
5.7
10
5.3
MTH 252 Midterm Review 10. Express the integral as a limit of Riemann sums. Do not evaluate the limit.
!
!+1
!"
!
! ! −2
11. Express the integral as a limit of Riemann sums. Do not evaluate the limit.
!"
(! ! + ! ! ) !"
!
12. Evaluate the definite integral.
!
(2! − ! ! ) !"
!!
13. Evaluate the definite integral.
!
2 ! !"
!
14. Evaluate the definite integral.
!
3! ! ! + 4 !"
!
15. Evaluate the definite integral.
!
!
sec !
!
!"
4
16. Use the graph provided to evaluate the given definite integrals.
! = !(!) i)
ii)
iii)
!
!(!) !"
!
!
! ! !"
!
!"
!(!) !"
!
MTH 252 Midterm Review 17. Find the exact area under the curve ! ! = 5! ! + 3 sin ! from ! = 0 to ! = !.
18. Find the exact area under the curve ! ! = 5! − 5! from ! = 0 to ! = 1.
19. Find the exact area under the curve ! ! = 3! − 1
!"
from ! = 1 to ! = 2.
20. Water flows from the bottom of a storage tank at a rate of ! ! = 200 − 4! liters per minute.
Find the amount of water that flows from the tank during the first ten minutes.
21. A particle moves along a line so that its velocity at time ! is ! ! = ! ! − ! − 6 (measured in
meters per second). Find both the displacement of the particle and distance traveled in the first
5 seconds.
22. Evaluate the indefinite integral.
1 !
− !"
! 2
23. Evaluate the indefinite integral.
!
4! ! − ! + 1 !"
24. Evaluate the indefinite integral.
!"
5+!
25. Evaluate the indefinite integral.
26. Evaluate the definite integral.
!
3!
!"
3! + 1
!
3! ! ! + 4 !"
!
27. Evaluate the definite integral.
!
sec !
!
!
!"
4
28. Find the area enclosed by the curves ! = ! ! , ! = ! ! − 1, ! = −1, and ! = 1.
!
29. Find the area enclosed by the curves ! = sin !, ! = !, ! = ! , and ! = !.
!
!
30. Find the area enclosed by the curves ! = !, ! = ! ! , and ! = 2.
31. Find the area enclosed by the curves 4! + ! ! = 12 and ! = !.
MTH 252 Midterm Review 32. Find the area enclosed by the curves ! = ! and ! = ! ! − 2.
33. Find the area enclosed by the curves ! + ! = 0 and ! = ! ! + 3!.
34. Find the volume of the solid obtained by rotating the region bounded by the curves ! = 2! and
! = ! ! about the !-axis.
35. Find the volume of the solid obtained by rotating the region bounded by the curves ! = 1 + ! !
and ! = ! − 3 about the !-axis.
36. Find the volume of the solid obtained by rotating the region bounded in the first quadrant by the
curves ! = ! ! and ! = 2! − ! ! about the !-axis.
37. Find the volume of the solid obtained by rotating the region bounded in the first quadrant by the
curves ! = ! ! and ! = 2! − ! ! about the !-axis.
38. Describe the solid represented by the integral.
!
!
2!" cos ! !"
!
39. Describe the solid represented by the integral
!
!
2! cos ! ! !"
!
40. Find the average value of the function ! ! =
!
41. Find the average value of the function ! ! =
! − 2! on the interval 1,8 .
!
! ! !!
on the interval 0,2 .
42. Find the average value of the function ! = !(!) on 0,8 .
! = !(!) MTH 252 Midterm Review 43. Find the values of ! such that the average value of ! ! = 2 + 6! − 3! ! on the interval 0, ! is
equal to 3.
44. In a certain city the temperature (in °F) ! hours after 9 AM was modeled by the function
!!
! ! = 50 + 14 sin
12
Find the average temperature during the period from 9 AM to 9 PM.