James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

436 Chapter 6 Applications of Integration
;
(b) Let P 3 be the point on the graph of t where infectiousness
begins. It has been shown that infectiousness ends
at a point P 4 on the graph of t where the line through
P 3, P 4 has the same slope as the line through P 1, P 2 in
Example 5(b). On what day does infectiousness end?
(c) Compute the level of infectiousness for this patient.
52. The rates at which rain fell, in inches per hour, in two
different locations t hours after the start of a storm
are given by f std − 0.73t 3 2 2t 2 1 t 1 0.6 and
tstd − 0.17t 2 2 0.5t 1 1.1. Compute the area between
the graphs for 0 < t < 2 and interpret your result in this
context.
53. Two cars, A and B, start side by side and accelerate from
rest. The figure shows the graphs of their velocity functions.
(a) Which car is ahead after one minute? Explain.
(b) What is the meaning of the area of the shaded region?
(c) Which car is ahead after two minutes? Explain.
(d) Estimate the time at which the cars are again side by
side.
√
0
A
B
1
2
t (min)
54. The figure shows graphs of the marginal revenue function
R9 and the marginal cost function C9 for a manufacturer.
[Recall from Section 4.7 that Rsxd and Csxd represent the
revenue and cost when x units are manufactured. Assume
that R and C are measured in thousands of dollars.] What
;
is the meaning of the area of the shaded region? Use the
Midpoint Rule to estimate the value of this quantity.
y
Rª(x)
3
2
1
Cª(x)
0 50 100
55. The curve with equation y 2 − x 2 sx 1 3d is called Tschirnhausen’s
cubic. If you graph this curve you will see that
part of the curve forms a loop. Find the area enclosed by the
loop.
56. Find the area of the region bounded by the parabola y − x 2 ,
the tangent line to this parabola at s1, 1d, and the x-axis.
57. Find the number b such that the line y − b divides the
region bounded by the curves y − x 2 and y − 4 into two
regions with equal area.
58. (a) Find the number a such that the line x − a bisects the
area under the curve y − 1yx 2 , 1 < x < 4.
(b) Find the number b such that the line y − b bisects the
area in part (a).
59. Find the values of c such that the area of the region bounded
by the parabolas y − x 2 2 c 2 and y − c 2 2 x 2 is 576.
60. Suppose that 0 , c , y2. For what value of c is the
area of the region enclosed by the curves y − cos x,
y − cossx 2 cd, and x − 0 equal to the area of the region
enclosed by the curves y − cossx 2 cd, x − , and y − 0?
61. For what values of m do the line y − mx and the curve
y − xysx 2 1 1d enclose a region? Find the area of the region.
x
APPLIED Project
The gini index
How is it possible to measure the distribution of income among the inhabitants of a given
country? One such measure is the Gini index, named after the Italian economist Corrado Gini,
who first devised it in 1912.
We first rank all households in a country by income and then we compute the percentage
of households whose income is at most a given percentage of the country’s total income. We
define a Lorenz curve y − Lsxd on the interval f0, 1g by plotting the point say100, by100d
on the curve if the bottom a% of households receive at most b% of the total income. For
instance, in Figure 1 (on page 437) the point s0.4, 0.12d is on the Lorenz curve for the United
States in 2010 because the poorest 40% of the population received just 12% of the total
income. Likewise, the bottom 80% of the population received 50% of the total income, so
the point s0.8, 0.5d lies on the Lorenz curve. (The Lorenz curve is named after the American
economist Max Lorenz.)
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