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Mathematics LibreTexts

7.6E: Exercises for Section 7.6

  • Gilbert Strang & Edwin “Jed” Herman
  • OpenStax

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In exercises 1 - 5, approximate the following integrals using either the midpoint rule, trapezoidal rule, or Simpson’s rule as indicated. (Round answers to three decimal places.)

1) 21dxx; trapezoidal rule; n=5

Answer
0.696

2) 304+x3dx; trapezoidal rule; n=6

3) 304+x3dx; Simpson’s rule; n=6

Answer
9.279

4) 120x2dx; midpoint rule; n=6

5) 10sin2(πx)dx; midpoint rule; n=3

Answer
0.500

6) Use the midpoint rule with eight subdivisions to estimate 42x2dx.

7) Use the trapezoidal rule with four subdivisions to estimate 42x2dx.

Answer
T4=18.75

8) Find the exact value of 42x2dx. Find the error of approximation between the exact value and the value calculated using the trapezoidal rule with four subdivisions. Draw a graph to illustrate.

Approximate the integral to four decimal places using the indicated rule.

9) 10sin2(πx)dx; trapezoidal rule; n=6

Answer
0.5000

10) 3011+x3dx; trapezoidal rule; n=6

11) 3011+x3dx; Simpson’s rule; n=6

Answer
1.1614

12) 0.80ex2dx; trapezoidal rule; n=4

13) 0.80ex2dx; Simpson’s rule; n=4

Answer
0.6577

14) 0.40sin(x2)dx; trapezoidal rule; n=4

15) 0.40sin(x2)dx; Simpson’s rule; n=4

Answer
0.0213

16) 0.50.1cosxxdx; trapezoidal rule; n=4

17) 0.50.1cosxxdx; Simpson’s rule; n=4

Answer
1.5629

18) Evaluate 10dx1+x2 exactly and show that the result is π/4. Then, find the approximate value of the integral using the trapezoidal rule with n=4 subdivisions. Use the result to approximate the value of π.

19) Approximate 421lnxdx using the midpoint rule with four subdivisions to four decimal places.

Answer
1.9133

20) Approximate 421lnxdx using the trapezoidal rule with eight subdivisions to four decimal places.

21) Use the trapezoidal rule with four subdivisions to estimate 0.80x3dx to four decimal places.

Answer
T(4)=0.1088

22) Use the trapezoidal rule with four subdivisions to estimate 0.80x3dx. Compare this value with the exact value and find the error estimate.

23) Using Simpson’s rule with four subdivisions, find π/20cos(x)dx.

Answer
π/20cos(x)dx1.0

24) Show that the exact value of 10xexdx=12e. Find the absolute error if you approximate the integral using the midpoint rule with 16 subdivisions.

25) Given 10xexdx=12e, use the trapezoidal rule with 16 subdivisions to approximate the integral and find the absolute error.

Answer
Approximate error is 0.000325.

26) Find an upper bound for the error in estimating 30(5x+4)dx using the trapezoidal rule with six steps.

27) Find an upper bound for the error in estimating 541(x1)2dx using the trapezoidal rule with seven subdivisions.

Answer
17938

28) Find an upper bound for the error in estimating 30(6x21)dx using Simpson’s rule with n=10 steps.

29) Find an upper bound for the error in estimating 521x1dx using Simpson’s rule with n=10 steps.

Answer
8125,000

30) Find an upper bound for the error in estimating π02xcos(x)dx using Simpson’s rule with four steps.

31) Estimate the minimum number of subintervals needed to approximate the integral 41(5x2+8)dx with an error magnitude of less than 0.0001 using the trapezoidal rule.

Answer
475

32) Determine a value of n such that the trapezoidal rule will approximate 101+x2dx with an error of no more than 0.01.

33) Estimate the minimum number of subintervals needed to approximate the integral 32(2x3+4x)dx with an error of magnitude less than 0.0001 using the trapezoidal rule.

Answer
174

34) Estimate the minimum number of subintervals needed to approximate the integral 431(x1)2dx with an error magnitude of less than 0.0001 using the trapezoidal rule.

35) Use Simpson’s rule with four subdivisions to approximate the area under the probability density function y=12πex2/2 from x=0 to x=0.4.

Answer
0.1554

36) Use Simpson’s rule with n=14 to approximate (to three decimal places) the area of the region bounded by the graphs of y=0,x=0, and x=π/2.

37) The length of one arch of the curve y=3sin(2x) is given by L=π/201+36cos2(2x)dx. Estimate L using the trapezoidal rule with n=6.

Answer
6.2807

38) The length of the ellipse x=acos(t),y=bsin(t),0t2π is given by L=4aπ/201e2cos2(t)dt, where e is the eccentricity of the ellipse. Use Simpson’s rule with n=6 subdivisions to estimate the length of the ellipse when a=2 and e=1/3.

39) Estimate the area of the surface generated by revolving the curve y=cos(2x),0xπ4 about the x-axis. Use the trapezoidal rule with six subdivisions.

Answer
4.606

40) Estimate the area of the surface generated by revolving the curve y=2x2,0x3 about the x-axis. Use Simpson’s rule with n=6.

41) The growth rate of a certain tree (in feet) is given by y=2t+1+et2/2, where t is time in years. Estimate the growth of the tree through the end of the second year by using Simpson’s rule, using two subintervals. (Round the answer to the nearest hundredth.)

Answer
3.41 ft

42) [T] Use a calculator to approximate 10sin(πx)dx using the midpoint rule with 25 subdivisions. Compute the relative error of approximation.

43) [T] Given 51(3x22x)dx=100, approximate the value of this integral using the midpoint rule with 16 subdivisions and determine the absolute error.

Answer
T16=100.125; absolute error = 0.125

44) Given that we know the Fundamental Theorem of Calculus, why would we want to develop numerical methods for definite integrals?

45) The table represents the coordinates (x,​y) that give the boundary of a lot. The units of measurement are meters. Use the trapezoidal rule to estimate the number of square meters of land that is in this lot.

x y x y
0 125 600 95
100 125 700 88
200 120 800 75
300 112 900 35
400 90 1000 0
500 90    
Answer
about 89,250 m2

46) Choose the correct answer. When Simpson’s rule is used to approximate the definite integral, it is necessary that the number of partitions be____

a. an even number

b. odd number

c. either an even or an odd number

d. a multiple of 4

47) The “Simpson” sum is based on the area under a ____.

Answer
parabola

48) The error formula for Simpson’s rule depends on___.

a. f(x)

b. f′(x)

c. f^{(4)}(x)

d. the number of steps


This page titled 7.6E: Exercises for Section 7.6 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin “Jed” Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform.

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