5.3 Riemann Sums and Definite Integrals 312
a.
b.
c.
d.
e.
____ 14. Sketch the region whose area is given by the definite integral and then use a
geometric formula to evaluate the integral.
a.
b.
c.
d.
e.
____ 15. Evaluate the integral.
given,
313 Chapter 5: Integration
–156
156
1,016
24
6,630
____ 16. Evaluate the integral.
given,
–180
16
5.3 Riemann Sums and Definite Integrals 314
–10169
–11522
–179
____ 17. Evaluate the integral.
given,
84
1074
8
1346
295
315 Chapter 5: Integration
____ 18. Evaluate the integral.
given,
–65
–2377
–66
–2024
–2025
5.3 Riemann Sums and Definite Integrals
316
____ 19.
The graph of f consists of line segments, as shown in the figure. Evaluate the definite
integral
using geometric formulas.
4
3
1
2
5
317 Chapter 5: Integration
5.3 Riemann Sums and Definite Integrals
Answer Section
5.4 The Fundamental Theorem of Calculus
318
5.4 The Fundamental Theorem of Calculus
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Evaluate the definite integral of the algebraic function.
Use a graphing utility to verify your results.
148.5
12
79.5
159
15
____ 2. Evaluate the definite integral . Use a graphing utility to verify your results.
a.
b.
c.
d.
e.
____ 3. Evaluate the definite integral of a function . Use a graphing utility to verify
your results.
a.
b.
c.
319 Chapter 5: Integration
d.
e.
____ 4. Evaluate the definite integral of the algebraic function.
Use a graphing utility to verify your results.
a.
b.
c.
d.
e.
____ 5. Evaluate the definite integral of the algebraic function.
Use a graphing utility to verify your results.
100
–30
196
–99
98
5.4 The Fundamental Theorem of Calculus
320
____ 6. Evaluate the definite integral of the algebraic function.
Use a graphing utility to verify your results.
–516
516
618
1185
–1974
____ 7. Evaluate the definite integral of the trig function.
Use a graphing utility to verify your results.
–7
–5
–6
–8
–4
____ 8. Evaluate the definite integral of the function.
Use a graphing utility to verify your results.
a.
b.
c.
d.
e.
321 Chapter 5: Integration
____ 9. Determine the area of the given region.
a.
b.
c.
d.
e.
____ 10. Determine the area of the given region.
5.4 The Fundamental Theorem of Calculus
322
a.
b.
c.
d.
e.
____ 11. Find the area of the region bounded by the graphs of the equations
. Round your answer to the nearest whole number.
768
96
49
72
16
____ 12. Find the average value of the function over the interval .
–152
–52
148
248
–252
____
13.
Find the average value of
on the interval
.
a.
b.
35
c.
d.
e.
40
____
14.
The force
(in newtons) of a hydraulic cylinder in a press is proportional to the
square of
where
is the distance (in meters) that the cylinder is extended in its cycle. The
domain of
is
and
. Find
as a function of .
a.
b.
c.
d.
e.
323
Chapter 5: Integration
____
15.
The force
(in newtons) of a hydraulic cylinder in a press is proportional to the
square of
where is the distance (in meters) that the cylinder is extended in its cycle. The
domain of
is
and
. Find the average force exerted by the press over the
interval
. Round your answer to the nearest whole number.
221 newtons
110 newtons
540 newtons
180 newtons
662 newtons
____ 16. The volume in liters of air in the lungs during a five-second respiratory cycle is
approximated by the model where is the time in seconds.
Approximate the average volume of air in the lungs during one cycle. Round your answer to
four decimal places.
1.8776 liter
2.0094 liter
0.4004 liter
0.8148 liter
0.5220 liter
____ 17. Find given
.
a.
b.
c.
d.
e.
____ 18. Find given
.
a.
b.
c.
d.
e.
5.4 The Fundamental Theorem of Calculus
324
5.4 The Fundamental Theorem of Calculus
Answer Section
1.
3.
6.
10.
12.
13.
14.
15.
18.
325 Chapter 5: Integration
5.5 Integration by Substitution
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1.
Find the indefinite integral
.
a.
b.
c.
d.
e.
____
2.
Find the indefinite integral
.
a.
b.
c.
d.
e.
5.5 Integration by Substitution
326
____
3.
Find the indefinite integral
.
a.
b.
c.
d.
e.
____
4.
Find the indefinite integral of the following function and check the result by
differentiation.
a.
b.
c.
d.
e.
327
Chapter 5: Integration
____
5.
Find the indefinite integral of the following function and check the result by
differentiation.
a.
b.
c.
d.
e.
____ 6. Find the indefinite integral of the following function and check the result by
differentiation.
a.
b.
c.
d.
e.
5.5 Integration by Substitution
328
____ 7. Solve the differential equation.
a.
b.
c.
d.
e.
____ 8. Solve the differential equation.
a.
b.
c.
d.
e.
329 Chapter 5: Integration
____ 9. Find the indefinite integral of the following function.
a.
b.
c.
d.
e.
____ 10. Find the indefinite integral .
a.
b.
c.
d.
e.
____ 11. Find the indefinite integral of the following function.
a.
b.
c. + C
d.
e.
5.5 Integration by Substitution
330
____ 12. Find the indefinite integral.
a.
b.
c.
d.
e.
____ 13. Find the indefinite integral.
a.
b.
c.
d.
e.
____ 14.
Find the indefinite integral
a.
b.
1
c.
32
1
d.
32
e.
1
32
331
Chapter 5: Integration
____
15.
Find the indefinite integral.
a.
b.
c.
d.
e.
____ 16. Find the indefinite integral of the following function.
a.
b.
c.
d.
e.
____ 17. Find the indefinite integral.
a.
b.
c.
d.
e.