Back45–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.
46. ∫(0 to 2) x⁴ dx; n = 30
c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.
45–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.
47. ∫(1 to e) (1/x) dx; n = 50
c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.
45–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.
48. ∫(0 to π/4) (1/(1 + x²)) dx; n = 64
c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.
49–52. {Use of Tech} Simpson’s Rule
Apply Simpson’s Rule to the following integrals. It is easiest to obtain the Simpson’s Rule approximations from the Trapezoid Rule approximations, as in Example 8. Make a table similar to Table 8.8 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.
51. ∫(from 0 to π) e⁻ᵗ sin(t) dt = ½(e⁻ᵖⁱ + 1)
54–57. {Use of Tech} Comparing the Midpoint and Trapezoid Rules Compare the errors in the Midpoint and Trapezoid Rules with n = 4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given).
54. ∫(from 0 to π/2) sin⁶x dx = 5π/32
54–57. {Use of Tech} Comparing the Midpoint and Trapezoid Rules Compare the errors in the Midpoint and Trapezoid Rules with n = 4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given).
59. ∫(from 0 to π) ln(5 + 3cosx) dx = π ln(9/2)
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. Suppose ∫_a^b f(x) dx is approximated with Simpson’s Rule using n = 18 subintervals, where |f^(4)(x)| ≤ 1 on [a, b]. The absolute error E_S in approximating the integral satisfies E_S ≤ (Δx)^5 / 10.
58–61. {Use of Tech} Using Simpson's Rule Approximate the following integrals using Simpson's Rule. Experiment with values of n to ensure the error is less than 10⁻³.
60. ∫(from 0 to π) ln(2 + cos x) dx = π ln((2 + √3)/2)
63. (Use of Tech) Normal distribution of heights
The heights of U.S. men are normally distributed with a mean of 69 in and a standard deviation of 3 in. This means that the fraction of men with a height between a and b (with a < b) inches is given by the integral
(1/(3√(2π))) ∫ₐᵇ e^(-((x-69)/3)²/2) dx.
What percentage of American men are between 66 and 72 inches tall? Use the method of your choice, and experiment with the number of subintervals until you obtain successive approximations that differ by less than 10⁻³.
64. (Use of Tech) Normal distribution of movie lengths
A study revealed that the lengths of U.S. movies are normally distributed with a mean of 110 minutes and a standard deviation of 22 minutes. This means that the fraction of movies with lengths between a and b minutes (with a < b) is given by the integral:
(1/(22√(2π))) ∫[a to b] e^(-((x-110)/22)²/2) dx.
What percentage of U.S. movies are between 1 hr and 1.5 hr long (60-90 min)?
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
66. Let f(x) = cos(x²).
a. Find a Midpoint Rule approximation to ∫[-1 to 1] cos(x²) dx using n = 30 subintervals.
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
66. Let f(x) = cos(x²).
d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
67. Let f(x) = √(x³ + 1).
a. Find a Midpoint Rule approximation to ∫[1 to 6] √(x³ + 1) dx using n = 50 subintervals.
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
67. Let f(x) = √(x³ + 1).
d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
68. Let f(x) = e^(x²).
a. Find a Trapezoid Rule approximation to ∫[0 to 1] e^(x²) dx using n = 50 subintervals.