Calculus 2nd Edition Rogawski

476 CHAPTER 8 TECHNIQUES OF INTEGRATION
69. Compute p(X ≤ 1), where X is a continuous random variable
1
with probability density p(x) =
π(x 2 + 1) .
70. Show that p(x) = 1 4 e−t/2 + 1 6 e−t/3 is a probability density and
find its mean.
71. Find a constant C such that p(x) = Cx 3 e −x2 is a probability
density and compute p(0 ≤ X ≤ 1).
72. The interval between patient arrivals in an emergency room
is a random variable with exponential density function p(x) =
0.125e −0.125t (t in minutes). What is the average time between patient
arrivals? What is the probability of two patients arriving within 3
minutes of each other?
73. Calculate the following probabilities, assuming that X is normally
distributed with mean µ = 40 and σ = 5.
(a) p(X ≥ 45) (b) p(0 ≤ X ≤ 40)
74. According to kinetic theory, the molecules of ordinary matter are
in constant random motion. The energy E of a molecule is a random
variable with density function p(E) = kT 1 e−E/(kT ) , where T is the
temperature (in kelvins) and k is Boltzmann’s constant. Compute the
mean kinetic energy E in terms of k and T .
In Exercises 75–84, determine whether the improper integral converges
and, if so, evaluate it.
∫ ∞
∫
dx
∞ dx
75.
0 (x + 2) 2 76.
4 x 2/3
77.
∫ 4
∫
dx
∞ dx
0 x 2/3 78.
9 x 12/5
79.
∫ 0
∫
dx
9
−∞ x 2 80. e 4x dx
+ 1
−∞
81.
∫ π/2
∫ ∞ dx
cot θ dθ 82.
0
1 (x + 2)(2x + 3)
∫ ∞
∫ 5
83. (5 + x) −1/3 dx 84. (5 − x) −1/3 dx
0
2
In Exercises 85–90, use the Comparison Test to determine whether the
improper integral converges or diverges.
∫ ∞
∫
dx
∞
85.
8 x 2 86. (sin 2 x)e −x dx
− 4
8
∫ ∞
∫
dx
∞ dx
87.
3 x 4 + cos 2 88.
x
1 x 1/3 + x 2/3
∫ 1
∫
dx
∞
89.
0 x 1/3 + x 2/3 90. e −x3 dx
0
91. Calculate the volume of the infinite solid obtained by rotating the
region under y = (x 2 + 1) −2 for 0 ≤ xα.
∫ 5
96. Estimate f (x) dx by computing T 2 , M 3 , T 6 , and S 6 for a
2
function f (x) taking on the values in the following table:
x 2 2.5 3 3.5 4 4.5 5
f (x)
1
2 2 1 0 − 3 2 −4 −2
97. State whether the approximation M N or T N is larger or smaller
than the integral.
∫ π
∫ 2π
(a) sin xdx
(b) sin xdx
0
π
∫ 8
∫
dx
5
(c)
1 x 2
(d) ln xdx
2
98. The rainfall rate (in inches per hour) was measured hourly during
a 10-hour thunderstorm with the following results:
0, 0.41, 0.49, 0.32, 0.3, 0.23,
0.09, 0.08, 0.05, 0.11, 0.12
Use Simpson’s Rule to estimate the total rainfall during the 10-hour
period.
In Exercises 99–104, compute the given approximation to the integral.
∫ 1
∫ 4 √
99. e −x2 dx, M 5 100. 6t 3 + 1 dt, T 3
0
2
∫ π/2 √
∫ 4 dx
101. sin θ dθ, M4 102.
π/4
1 x 3 + 1 , T 6
∫ 1
∫ 9
103. e −x2 dx, S 4 104. cos(x 2 )dx, S 8
0
5
105. The following table gives the area A(h) of a horizontal cross section
of a pond at depth h. Use the Trapezoidal Rule to estimate the
volume V of the pond (Figure 1).
h (ft) A(h) (acres) h (ft) A(h) (acres)
0 2.8 10 0.8
2 2.4 12 0.6
4 1.8 14 0.2
6 1.5 16 0.1
8 1.2 18 0