272 Chapter 4: Integration
4.6 Numerical Integration
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Use the Trapezoid Rule to approximate the value of the definite integral with
. Round your answer to four decimal places.
a.
b.
c.
d.
e.
____ 2. Use Simpson’s Rule to approximate the value of the definite integral with
. Round your answer to four decimal places.
a.
b.
c.
d.
e.
____ 3. Apply the Trapezoidal Rule and Simpson’s Rule to approximate the value of the
definite integral using subintervals. Round your answer to six decimal places and compare the result
with the exact value of the definite integral.
a. The Trapezoidal rule gives and Simpson’s rule gives .
b. The Trapezoidal rule gives and Simpson’s rule gives .
c. The Trapezoidal rule gives and Simpson’s rule gives .
d. The Trapezoidal rule gives 1.366795 and Simpson’s rule gives .
e. The Trapezoidal rule gives 1.366795 and Simpson’s rule gives .
____ 4. Apply the Trapezoidal Rule and Simpson’s Rule to approximate the value of the
definite integral using subintervals. Round your answer to six decimal places and compare the
result with the exact value of the definite integral.
4.6 Numerical Integration 273
a. The Trapezoidal rule gives 16.3400 and Simpson’s rule gives .
b. The Trapezoidal rule gives 16.3400 and Simpson’s rule gives .
c. The Trapezoidal rule gives and Simpson’s rule gives .
d. The Trapezoidal rule gives and Simpson’s rule gives .
e. The Trapezoidal rule gives and Simpson’s rule gives .
____ 5. Apply the Trapezoidal Rule and Simpson’s Rule to approximate the value of the
definite integral using subintervals. Round your answer to six decimal places and compare the result
with the exact value of the definite integral.
a. The Trapezoidal rule gives 43.377044 and Simpson’s rule gives .
b. The Trapezoidal rule gives and Simpson’s rule gives .
c. The Trapezoidal rule gives 43.377044 and Simpson’s rule gives .
d. The Trapezoidal rule gives and Simpson’s rule gives .
e. The Trapezoidal rule gives and Simpson’s rule gives .
____ 6. Apply the Trapezoidal Rule and Simpson’s Rule to approximate the value of the
definite integral using subintervals. Round your answer to six decimal places and compare the result
with the exact value of the definite integral.
a. The Trapezoidal rule gives and Simpson’s rule gives .
b. The Trapezoidal rule gives 1.3973 and Simpson’s rule gives .
c. The Trapezoidal rule gives and Simpson’s rule gives .
d. The Trapezoidal rule gives 1.3973 and Simpson’s rule gives .
e. The Trapezoidal rule gives and Simpson’s rule gives .
____ 7. Use the Trapezoidal Rule to approximate the value of the definite integral
with . Round your answer to four decimal places.
a. –8.0654
b. –4.0327
c. –2.0164
d. –1.0082
e. –0.5041
274 Chapter 4: Integration
____ 8. Use Simpson’s Rule to approximate the value of the definite integral
with . Round your answer to four decimal places.
a. 2.9784
b. 3.0627
c. 8.0100
d. 10.6800
e. 2.6700
____ 9. Use the error formula to estimate the error in approximating the integral
with using Trapezoidal Rule.
a. 0
b. 9
c. 3
d. 6
e. 1
____ 10. Estimate the error in using (a) the Trapezoidal Rule and (b) Simpson’s Rule with
when approximating the following integral.
a. The error for the Trapezoidal Rule is 0.0051 and for Simpson’s Rule it is 0.0013.
b. The error for the Trapezoidal Rule is 0.0000 and for Simpson’s Rule it is 0.0200.
c. The error for the Trapezoidal Rule is 0.0204 and for Simpson’s Rule it is 0.0200.
d. The error for the Trapezoidal Rule is 0.0204 and for Simpson’s Rule it is 0.0000.
e. The error for the Trapezoidal Rule is 0.0000 and for Simpson’s Rule it is 0.0000.
____ 11. Use the error formula to estimate the error in approximating the integral
with using Simpson’s Rule. Round your answer to six decimal places.
a. 0.007084
b. 0.141676
c. 0.004723
d. 0.003936
e. 0.002530
4.6 Numerical Integration 275
____ 12. Find the smallest n such that the error estimate from the error formula in the
approximation of the definite integral is less than 0.00001 using the Trapezoidal Rule.
a. 49
b. 26
c. 73
d. 30
e. 15
____ 13. Find the smallest n such that the error estimate in the approximation of the definite
integral is less than 0.00001 using Simpson’s Rule.
a. 48
b. 8
c. 11
d. 14
e. 19
____ 14. Find the smallest n such that the error estimate in the approximation of the definite
integral is less than 0.00001 using Simpson’s Rule.
a. 10
b. 16
c. 6
d. 8
e. 13
____ 15. Find the smallest n such that the error estimate in the approximation of the definite
integral is less than 0.00001 using the Trapezoidal Rule. Use a graphing utility to
estimate the maximum of the absolute value of the second derivative.
a. 208
b. 189
c. 196
d. 201
e. 188
276 Chapter 4: Integration
____ 16. To determine the size of the motor required to operate a press, a company must know
the amount of work done when the press moves an object linearly 5 feet. Suppose that the variable
force to move the object is , where F is given in pounds, and x gives the
position of the unit in feet. Use Simpson’s Rule with to approximate the work W (in foot-
pounds) done through one cycle . Round your answer to nearest unit.
a. 98,336 foot-pounds
b. 106,541 foot-pounds
c. 105,336 foot-pounds
d. 100,131 foot-pounds
e. 102,336 foot-pounds
____ 17. The table lists several measurements gathered in an experiment to approximate an
unknown continuous function . Approximate the integral using the Trapezoidal
Rule. Round your answer to three decimal places.
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
4.12 4.27 4.62 6.37 6.82 7.47 7.72 8.67 8.92
a. 13.115
b. 12.000
c. 7.373
d. 104.920
e. 24.600
____ 18. The table lists several measurements gathered in an experiment to approximate an
unknown continuous function . Approximate the integral using the Simpson’s Rule.
Round your answer to three decimal places.
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
4.72 4.77 4.82 6.17 6.52 7.17 7.32 7.97 8.02
a. 12.197
b. 8.518
c. 15.975
d. 12.865
e. 11.628
4.6 Numerical Integration 277
____ 19. Use the Trapezoidal Rule to estimate the number of square meters of land in a lot
where x and y are measured in meters as shown in the figure below. The land is bounded by a stream
and two straight roads that meet at right angles.
0 100 200 300 400 500 600 700 800 900 1000
110 110 105 97 75 75 80 65 60 20 0
a. 74,200
b. 74,000
c. 73,700
d. 74,700
e. 74,400
____ 20. Use a computer algebra system and Simpson’s Rule with to approximate t in
the integral equation . Round your answer to three decimal places.
a. 3.501
b. 3.581
c. 3.901
d. 3.529
e. 3.171
____ 21. Use Simpson’s Rule with to approximate using the equation
. Round your answer to five decimal places.
a. 3.14159
b. 3.13595
c. 3.14723
d. 3.14381
e. 3.13937
278 Chapter 4: Integration
4.6 Numerical Integration
Answer Section
4.6 Numerical Integration 279