Name: ________________________ Class: ___________________ Date: __________
ID: A
Test 4 Review
Short Answer
1. Find the general solution of the differential
equation below and check the result by
differentiation.
dY 9
 u
du 4

7. Find the indefinite integral x 2
3  x 3 dx .
8. Find the indefinite integral of the following
function.
5
4
  sin5z dz
2. Use the error formula to estimate the error in
6
Ê
ˆ5
9. Find the indefinite integral t 3 ÁÁÁ 5  t 4 ˜˜˜ dt .
Ë
¯


approximating the integral (9x  1) dx with n  4
3
using Trapezoidal Rule.
Ê
ˆ
5 ÁÁ x 2  5 ˜˜
Ë
¯
10. Find the average value of f (x ) 
on the
2
x
È ˘
interval ÍÍÎ 1,3˙˙˚ .
3. Find the smallest n such that the error estimate in
the approximation of the definite integral
1
 2cos(x) dx is less than 0.00001 using Simpson's
11. Find the average value of the function over the
given interval and all values t in the interval for
which the function equals its average value.
0
Rule.
4. Use the Trapezoid Rule to approximate the value of
f (t ) 
2

the definite integral x 4 dx wth n  4. Round your
0
Use a graphing utility to verify your results.
answer to four decimal places.
12. Find the indefinite integral
5. Find the indefinite integral of the following
function and check the result by differentiation.
 (5  s )
5

 (1  4z) 5 dz .
È ˘
13. Determine all values of x in the interval ÍÍÎ 1,3˙˙˚ for
Ê
ˆ
4 ÁÁ x 2  1 ˜˜
Ë
¯
equals its
which the function f (x ) 
2
x
16
average value .
3
s ds
6. Evaluate the following definite integral.
0
t2  4
,1t5
t2
Ê
ˆ
z cos ÁÁÁ 7z 2 ˜˜˜ dz
Ë
¯
14. Find the area of the region bounded by the graphs
of the equations y  x 3  x, x  4, y  0 . Round
your answer to the nearest whole number.
Use a graphing utility to check your answer.
1
Name: ________________________
ID: A
15. Determine the area of the given region.
1

3
2
19. Evaluate the definite integral of a function u du.
y  x  sinx, 0  x  
0
Use a graphing utility to verify your results.
20. Evaluate the integral.
4
ˆ
Ê
 ÁÁÁË 24x 2  1˜˜˜¯ d x
3
given,
4
 x 3 dx 
175
,
4
3
4
16. Evaluate the definite integral of the trig function.
5
0
 x 2 dx 
37
,
3
3
(3sinx  4 cos x)dx
4
 x dx 
Use a graphing utility to verify your results.
7
,
2
3
17. Evaluate the definite integral of the algebraic
function.
5
0
4
 dx 1 .
3
4 x  4 dx
21. Sketch the region whose area is given by the
definite integral and then use a geometric formula
to evaluate the integral.
Use a graphing utility to verify your results.
18. Evaluate the definite integral of the function.
a

5x
 (2t  5cos t ) dt
a 2  z2 d z
a
0
22. Write the following limit as a definite integral on
the interval [2 , 5], where ci is any point in the ith
subinterval.
Use a graphing utility to verify your results.
n
lim
 ÁÊË 7c i  2˜ˆ¯ x i
 x  0 i  1
2
Name: ________________________
ID: A
23. Write the following limit as a definite integral on
˘
È
the interval ÍÍÎ 2, 8 ˙˙˚ where ci is any point in the i th
subinterval.
27. Sketch the region whose area is given by the
definite integral and then use a geometric formula
to evaluate the integral.
1
n
lim

 x  0 i  1
 ÊÁË 1  u ˆ˜¯ d u
3c 2i  4c i x i
1
ÊÁ ˆ˜
ÁÁ 6 ˜˜
24. Write the limit lim  ÁÁÁÁ 2 ˜˜˜˜ x i , as a definite
   0 i  1 Á
ÁË c i ˜˜¯
˘
È
integral on the interval ÍÍÎ 8, 10 ˙˙˚ , where ci is any
point in the i th subinterval.
28. Evaluate the following definite integral by the limit
definition.
25. The graph of the function f (x )  16  x 2 is given
below. Which of the following definite integrals
yields the area of the shaded region?
29. Use the summation formulas to rewrite the
n
1
 5s 3 ds
2
n
expression
 24k (k7  1)
k1
n
without the summation
notation.
30. Find the limit.
n
Ê
ˆÊ ˆ
n   i  1Ë
¯Ë ¯
lim
˜˜ ÁÁ 8 ˜˜
˜Á ˜
 ÁÁÁÁÁ 10i
n ˜˜ ÁÁ n ˜˜
31. Use the limit process to find the area of the region
between the graph of the function y  16  x 2 and
˘
È
x-axis over the interval ÍÍÎ 4,4 ˙˙˚ .
32. Find the sum given below.
8
26. Evaluate the following definite integral by the limit
definition.
 k (k  4)
k3
4
Ê
ˆ
 ÁÁÁË 9s 2  4˜˜˜¯ ds
33. Use sigma notation to write the sum
5
5
5
5



.
1  22
11 12 13
2
3
Name: ________________________
ID: A
34. Solve the differential equation.
41. The diagram below shows upper and lower sums
for the function
dP
 15x 4  5, P (2)  8
dx
using 4 subintervals. Use upper and lower sums to
approximate the area of the region using the 4
subintervals.
35. Find the limit of s (n) as n  .
5
s (n )  3
n
ÈÍ
˘
ÍÍ n 3 (n  1) 2 ˙˙˙
ÍÍ
˙˙
ÍÍ
˙˙
ÍÍ
˙˙
7
ÍÍÎ
˙˙˚
36. The maker of an automobile advertises that it takes
12 seconds to accelerate from 30 kilometers per
hour to 85 kilometers per hour. Assuming constant
acceleration, compute the acceleration in meters per
second per second. Round your answer to three
decimal places.
37. A ball is thrown vertically upwards from a height
of 10 ft with an initial velocity of 30 ft per second.
How high will the ball go? Note that the
acceleration of the ball is given by a(t)  32 feet
per second per second.

42. Find the indefinite integral
 ÁÊÁË 12u
43. Find the indefinite integral
 ÊÁÁË 12s
38. Find the indefinite integral 10 sins  7cos s ds .
39. Find the indefinite integral and check the result by
differentiation.

3z 2  12z  9
dz
z4
40. Find the indefinite integral

13
x 8 dx.
4
2
3
ˆ
 4u ˜˜ du .
¯
ˆ
 12s  3 ˜˜ ds .
¯
ID: A
Test 4 Review
Answer Section
SHORT ANSWER
1. ANS:
9
4
Y (u )  u  C
PTS: 1
DIF: Medium
REF: 4.1.7
OBJ: Calculate the general solution of a differential equation
NOT: Section 4.1
2. ANS:
0
MSC: Skill
PTS: 1
DIF: Easy
REF: 4.6.24a
OBJ: Estimate the error in using the Trapezoidal Rule to approximate a definite integral
MSC: Skill
NOT: Section 4.6
3. ANS:
19
PTS: 1
DIF: Medium
REF: 4.6.33b
OBJ: Identify the smallest value of n needed to approximate a definite integral to within a desired degree of
accuracy
MSC: Skill
NOT: Section 4.6
4. ANS:
7.0625
PTS: 1
DIF: Easy
REF: 4.6.1
OBJ: Approximate a definite integral using the Trapezoidal Rule
MSC: Skill
NOT: Section 4.6
5. ANS:
3
5
10 2 2 2
s  s C
5
3
PTS: 1
DIF: Easy
REF: 4.5.37
OBJ: Evaluate the indefinite integral of a function
NOT: Section 4.5
6. ANS:
0
MSC: Skill
PTS: 1
DIF: Medium
REF: 4.5.85
OBJ: Evaluate the definite integral of a function using substitution
MSC: Skill
NOT: Section 4.5
1
ID: A
7. ANS:
3
Ê
ˆ
2 ÁÁÁ 3  x 3 ˜˜˜ 2
Ë
¯
C
9
PTS: 1
DIF: Medium
REF: 4.5.20
OBJ: Evaluate the indefinite integral of a function using substitution
MSC: Skill
NOT: Section 4.5
8. ANS:
cos5 z
C
5
PTS: 1
DIF: Easy
REF: 4.5.47
OBJ: Evaluate the indefinite integral of a function using substitution
MSC: Skill
NOT: Section 4.5
9. ANS:
ÊÁ
ˆ6
ÁÁ 5  t 4 ˜˜˜
Ë
¯
C
24
PTS: 1
DIF: Easy
REF: 4.5.15
OBJ: Evaluate the indefinite integral of a function using substitution
MSC: Skill
NOT: Section 4.5
10. ANS:
40
3
PTS: 1
DIF: Easy
REF: 4.4.52a
OBJ: Calculate the average value of a function over a given interval
MSC: Skill
NOT: Section 4.4
11. ANS:
9
The average is and the point at which the function is equal to its mean value is
5
5.
PTS: 1
DIF: Medium
REF: 4.4.52
OBJ: Calculate the average value of a function over a given interval and identify the point at which it occurs
MSC: Skill
NOT: Section 4.4
12. ANS:
6
(1  4z)
C
24
PTS: 1
DIF: Easy
REF: 4.5.11
OBJ: Evaluate the indefinite integral of a function using substitution
MSC: Skill
NOT: Section 4.5
2
ID: A
13. ANS:
x 3
PTS: 1
DIF: Easy
REF: 4.4.52b
OBJ: Identify the points where a function equals its average value over a given interval
MSC: Skill
NOT: Section 4.4
14. ANS:
72
PTS: 1
MSC: Application
15. ANS:
0.5 2  2
DIF: Medium
NOT: Section 4.4
REF: 4.4.40
OBJ: Calculate the area bounded by a function
PTS: 1
MSC: Application
16. ANS:
–6
DIF: Medium
NOT: Section 4.4
REF: 4.4.38
OBJ: Calculate the area bounded by a function
PTS: 1
DIF: Easy
REF: 4.4.27
OBJ: Evaluate the definite integral of a function
NOT: Section 4.4
17. ANS:
98
PTS: 1
DIF: Medium
REF: 4.4.23
OBJ: Evaluate the definite integral of a function
NOT: Section 4.4
18. ANS:
25 2
PTS: 1
DIF: Easy
REF: 4.4.34
OBJ: Evaluate the definite integral of a function
NOT: Section 4.4
19. ANS:
2
5
PTS: 1
DIF: Medium
REF: 4.4.16
OBJ: Evaluate the definite integral of a function
NOT: Section 4.4
20. ANS:
295
PTS: 1
DIF: Easy
REF: 4.3.38
OBJ: Evaluate the definite integral of a function
NOT: Section 4.3
3
MSC: Skill
MSC: Skill
MSC: Skill
MSC: Skill
MSC: Skill
ID: A
21. ANS:
 a2
2
PTS: 1
MSC: Skill
22. ANS:
DIF: Easy
NOT: Section 4.3
REF: 4.3.32
OBJ: Evaluate a definite integral geometrically
5
 (7x  2)dx
2
PTS: 1
DIF: Easy
REF: 4.3.9
OBJ: Evaluate a definite integral by the limit definition
NOT: Section 4.3
23. ANS:
MSC: Skill
8

3x 2  4x dx
2
PTS: 1
DIF: Easy
REF: 4.3.11
OBJ: Write a limit as a definite integral on an interval
NOT: Section 4.3
24. ANS:
MSC: Skill
10
 x 2 dx
6
8
PTS: 1
DIF: Easy
REF: 4.3.12
OBJ: Write a limit as a definite integral on an interval
NOT: Section 4.3
25. ANS:
MSC: Skill
4
Ê
ˆ
 ÁÁÁË 16  x 2 ˜˜˜¯ dx
4
PTS: 1
DIF: Easy
REF: 4.3.17
OBJ: Write a definite integral for a bounded region
NOT: Section 4.3
4
MSC: Skill
ID: A
26. ANS:
240
PTS: 1
DIF: Easy
REF: 4.3.8
OBJ: Evaluate a definite integral by the limit definition
NOT: Section 4.3
27. ANS:
1
PTS: 1
MSC: Skill
28. ANS:
75

4
DIF: Easy
NOT: Section 4.3
REF: 4.3.29
PTS: 1
DIF: Easy
REF: 4.3.5
OBJ: Evaluate a definite integral by the limit definition
NOT: Section 4.3
29. ANS:
8
8
 6
4
n
n
PTS: 1
DIF: Medium
REF: 4.2.47
OBJ: Rewrite a sum without summation notation
NOT: Section 4.2
30. ANS:
40
PTS: 1
MSC: Skill
31. ANS:
256
3
DIF: Medium
NOT: Section 4.2
REF: 4.2.50
MSC: Skill
OBJ: Evaluate a definite integral geometrically
MSC: Skill
MSC: Skill
OBJ: Evaluate an infinite limit of a sum
PTS: 1
DIF: Medium
REF: 4.2.62
OBJ: Calculate the area bounded by a function using the limiting process
MSC: Application NOT: Section 4.2
32. ANS:
67
PTS: 1
MSC: Skill
DIF: Easy
NOT: Section 4.2
REF: 4.2.2
5
OBJ: Calculate a sum given in sigma notation
ID: A
33. ANS:
22
 1 5 j
j1
PTS: 1
DIF: Easy
MSC: Skill
NOT: Section 4.2
34. ANS:
P (x )  3x 5  5x  94
REF: 4.2.8
OBJ: Write a sum in sigma notation
PTS: 1
MSC: Skill
35. ANS:
unbounded
DIF: Medium
NOT: Section 4.1
REF: 4.1.59
OBJ: Solve a differential equation
PTS: 1
MSC: Skill
36. ANS:
1.273 m/sec2
DIF: Medium
NOT: Section 4.2
REF: 4.2.37
OBJ: Evaluate limits at infinity
PTS: 1
MSC: Application
37. ANS:
24.0625 ft
DIF: Medium
NOT: Section 4.1
REF: 4.1.85a
OBJ: Calcuate the acceleration
PTS: 1
DIF: Medium
REF: 4.1.71
OBJ: Solve differential equations related to position/velocity/acceleration
MSC: Application NOT: Section 4.1
38. ANS:
10 cos s  7sin s  C
PTS: 1
DIF: Easy
REF: 4.1.35
OBJ: Evaluate the indefinite integral of a function
NOT: Section 4.1
39. ANS:
3 6
3
  2  3 C
z z
z
PTS: 1
DIF: Medium
REF: 4.1.28
OBJ: Evaluate the indefinite integral of a function
NOT: Section 4.1
40. ANS:
13 21 13
x
C
21
PTS: 1
DIF: Easy
REF: 4.1.23
OBJ: Evaluate the indefinite integral of a function
NOT: Section 4.1
6
MSC: Skill
MSC: Skill
MSC: Skill
ID: A
41. ANS:
lower: 1.166 ; upper: 1.666
PTS: 1
DIF: Medium
REF: 4.2.42
OBJ: Estimate the area of a region using upper and lower sums MSC: Skill
NOT: Section 4.2
42. ANS:
4u 3  2u 2  C
PTS: 1
DIF: Easy
REF: 4.1.17
OBJ: Evaluate the indefinite integral of a function
NOT: Section 4.1
43. ANS:
3s 4  6s 2  3s  C
PTS: 1
DIF: Easy
REF: 4.1.20
OBJ: Evaluate the indefinite integral of a function
NOT: Section 4.1
7
MSC: Skill
MSC: Skill