256 Chapter 4: Integration
4.4 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.
a. 148.5
b. 12
c. 79.5
d. 159
e. 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.
d.
e.
4.4 The Fundamental Theorem of Calculus 257
____ 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.
a. 100
b. –30
c. 196
d. –99
e. 98
____ 6. Evaluate the definite integral of the algebraic function.
Use a graphing utility to verify your results.
a. –516
b. 516
c. 618
258 Chapter 4: Integration
d. 1185
e. –1974
____ 7. Evaluate the definite integral of the function.
Use a graphing utility to verify your results.
a.
b.
c.
d.
e.
____ 8. Determine the area of the given region.
a.
b.
c.
d.
e.
4.4 The Fundamental Theorem of Calculus 259
____ 9. Determine the area of the given region.
a.
b.
c.
d.
e.
____ 10. Find the area of the region bounded by the graphs of the equations
. Round your answer to the nearest whole number.
a. 768
b. 96
c. 49
d. 72
e. 16
____ 11. Find the average value of the function over the interval .
a. –152
b. –52
c. 148
d. 248
e. –252
260 Chapter 4: Integration
____ 12. 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.
Use a graphing utility to verify your results.
a. The average is and the point at which the function is equal to its mean value is .
b. The average is and the point at which the function is equal to its mean value is and
.
c. The average is and the point at which the function is equal to its mean value is .
d. The average is and the point at which the function is equal to its mean value is .
e. The average is and the point at which the function is equal to its mean value is and
.
____ 13. Find the average value of on the interval .
a.
b. 35
c.
d.
e. 40
4.4 The Fundamental Theorem of Calculus 261
____ 14. Determine all values of in the interval for which the function
equals its average value .
a.
b.
c.
d.
e.
____ 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 as a function of .
a.
b.
c.
d.
e.
____ 16. 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.
a. 221 newtons
b. 110 newtons
c. 540 newtons
d. 180 newtons
e. 662 newtons
____ 17. 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.
a. 1.8776 liter
b. 2.0094 liter
c. 0.4004 liter
d. 0.8148 liter
e. 0.5220 liter
262 Chapter 4: Integration
____ 18. Find given
.
a.
b.
c.
d.
e.
____ 19. Find given
.
a.
b.
c.
d.
e.
____ 20. Find given
.
a.
b.
c.
d.
e.
4.4 The Fundamental Theorem of Calculus 263
4.4 The Fundamental Theorem of Calculus – Answer Section