5.1 Antiderivatives and Indefinite Integration
292
5.1 Antiderivatives and Indefinite Integration
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the general solution of the differential equation below and check the result by
differentiation.
a.
b.
c.
d.
e.
____ 2. Find the general solution of the differential equation below and check the result by
differentiation.
a.
b.
c.
d.
e.
____ 3. Find the indefinite integral .
a.
b.
c.
none of the above
293
Chapter 5: Integration
____
4.
Find the indefinite integral
.
a.
b.
c.
d.
e.
____
5.
Find the indefinite integral
.
a.
b.
c.
d.
e.
____
6.
Find the indefinite integral
.
a.
b.
c.
d.
e.
____
7.
Find the indefinite integral
.
a.
b.
c.
d.
e.
5.1 Antiderivatives and Indefinite Integration
294
____ 8.
Find the indefinite integral and check the result by differentiation.
a.
b.
c.
d.
e.
____ 9. Find the indefinite integral .
a.
b.
c.
d.
e.
____ 10. Find the indefinite integral .
a.
b.
c.
d.
e.
____ 11. Find the indefinite integral .
a.
b.
c.
d.
e.
295 Chapter 5: Integration
____ 12. Solve the differential equation.
a.
b.
c.
d.
e.
____ 13. Solve the differential equation.
a.
b.
c.
d.
e.
____ 14. Solve the differential equation.
a.
b.
c.
d.
e.
____ 15. A ball is thrown vertically upwards from a height of ft with an initial velocity of ft
per second. How high will the ball go? Note that the acceleration of the ball is given by
feet per second per second.
ft
ft
ft
ft
ft
5.1 Antiderivatives and Indefinite Integration
296
____ 16.
The height above the ground of an object thrown upward from a point
feet above
the ground with an initial velocity of
feet per second is given by the function
. A balloon, rising vertically with a velocity of
feet per second, releases a
sandbag at the instant it is feet above the ground. How many seconds after its release will the bag
strike the ground? Round your answer to three decimal places.
seconds
seconds
seconds
seconds
seconds
____ 17.
The height above the ground of an object thrown upward from a point
feet above
the ground with an initial velocity of
feet per second is given by the function
. A balloon, rising vertically with a velocity of feet per second, releases a
sandbag at the instant it is feet above the ground. At what velocity will it hit the ground? Round
your answer to three decimal places.
ft/sec
ft/sec
ft/sec
ft/sec
ft/sec
____ 18.
The maker of an automobile advertises that it takes
seconds to accelerate from
kilometers per hour to
kilometers per hour. Assuming constant acceleration, compute the
acceleration in meters per second per second. Round your answer to three decimal places.
m/sec2
m/sec2
m/sec2
m/sec2
m/sec2
____ 19.
The maker of an automobile advertises that it takes
seconds to accelerate from
kilometers per hour to
kilometers per hour. Assuming constant acceleration, compute the distance,
in meters, the car travels during the
seconds. Round your answer to two decimal places.
m
m
m
m
m
297 Chapter 5: Integration
5.1 Antiderivatives and Indefinite Integration
Answer Section
5.2 Area
298
5.2 Area
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the sum given below.
42
44
38
50
40
____ 2. Find the sum given below.
3
33
25
67
43
____ 3. Find the sum given below.
a.
b.
c.
d.
e.
299
Chapter 5: Integration
____
4.
Use sigma notation to write the sum
.
a.
b.
c.
d.
e.
____
5.
Use the properties of summation and Theorem 5.2 to evaluate the sum.
a.
667
b.
552
c.
575
d.
1311
e.
391
____
6.
Use the properties of summation and Theorem 5.2 to evaluate the sum.
1156
1190
2108
2244
323
5.2 Area
300
____ 7. Use the properties of summation and Theorem 5.2 to evaluate the sum.
9,176
10,471
18,907
9,139
98,346
____ 8. Use the properties of summation and Theorem 5.2 to evaluate the sum.
239
7,656
7,612
3,861
7,590
____ 9. Use the properties of summation and Theorem 5.2 to evaluate the sum.
2,156
64,262
68,057
258,060
66,033
____ 10. Use left endpoints and 10 rectangles to find the approximation of the area of the region
between the graph of the function and the x-axis over the interval . Round your
answer to the nearest integer.
2925
3325
3000
3250
3125
301 Chapter 5: Integration
____ 11. Use left endpoints and 12 rectangles to find the approximation of the area of the region
between the graph of the function and the x-axis over the interval . Round your answer
to four decimal places.
0.0419
0.1319
0.1219
0.0819
0.0119
____ 12. Find the limit of as .
5/7
5
1/7
10/7
unbounded
____ 13. Find the limit of as .
1/10
5
unbounded
1/2
1
5.2 Area
302
____ 14. The diagram below shows upper and lower sums for the function
using 4 subintervals. Use upper and lower sums to approximate the area of the region using the
4 subintervals.
lower: 1.166 ; upper: 1.666
lower: 4.664 ; upper: 6.664
lower: 2.332 ; upper: 3.332
lower: 0.916 ; upper: 1.916
____ 15. Use the summation formulas to rewrite the expression without the
summation notation.
a.
b.
c.
d.
e.
303 Chapter 5: Integration
____ 16. Find the limit.
20
18
36
80
40
____ 17. Use the limit process to find the area of the region between the graph of the function
and x-axis over the interval .
a.
b.
c.
d.
e.
5.2 Area 304
5.2 Area
Answer Section
305 Chapter 5: Integration
5.3 Riemann Sums and Definite Integrals
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Evaluate the following definite integral by the limit definition.
a.
b.
c.
d.
e.
____ 2. Evaluate the following definite integral by the limit definition.
4375
4372
1125
5.3 Riemann Sums and Definite Integrals 306
____ 3. Evaluate the following definite integral by the limit definition.
–216
240
–240
192
216
____ 4. Write the following limit as a definite integral on the interval [2 , 5], where ci is any point
in the ith subinterval.
a.
b.
c.
d.
e.
307
Chapter 5: Integration
____
5.
Write the limit
, as a definite integral on the interval
where
is any point in the
subinterval.
a.
b.
c.
d.
e.
____
6.
Write the following limit as a definite integral on the interval
where c
i
is any
point in the
subinterval.
a.
b.
c.
5.3 Riemann Sums and Definite Integrals 308
d.
e.
____ 7.
Write the limit
, as a definite integral on the interval
,
where ci is any point in the
subinterval.
a.
b.
c.
d.
e.
309
Chapter 5: Integration
____
8.
The graph of the function
is given below. Which of the following
definite integrals yields the area of the shaded region?
a.
b.
c.
d.
e.
5.3 Riemann Sums and Definite Integrals
310
____ 9.
The graph of the function
is given below. Which of the following
definite integrals yields the area of the shaded region?
a.
b.
c.
d.
e.
311
Chapter 5: Integration
____
10.
Sketch the region whose area is given by the definite integral and then use a
geometric formula to evaluate the integral.
30
15
4
34
–17
____ 11. Sketch the region whose area is given by the definite integral and then use a
geometric formula to evaluate the integral.
–158
12
80
82
–160
____ 12. Sketch the region whose area is given by the definite integral and then use a
geometric formula to evaluate the integral.
–1
1
2
4
15
____ 13. Sketch the region whose area is given by the definite integral and then use a
geometric formula to evaluate the integral.