# Mathematics

27–30. Two approaches Express the area of the following shaded regions in terms of (a) one or more integrals with respect to x and (b) one or more integrals with respect to y. You do not need to evaluate the integrals.

27.

x

y

y ! x

y ! x2 ” 2

28.

x

y

O

y ! x2 ” 4x

y ! 2x ” 8

29.

x

y

x ! 2y

x ! y2 ” 3

30.

x

y

O

y ! !x

y ! x3

31–32. Two approaches Find the area of the following regions by (a) integrating with respect to x and (b) integrating with respect to y. Be sure your results agree. Sketch the bounding curves and the region in question.

31. The region bounded by y = 2 – x 2

and x = 2y2

32. The region bounded by x = 2 – y2 and x = ! y !

33–38. Any method Use any method (including geometry) to find the area of the following regions. In each case, sketch the bounding curves and the region in question.

33. The region in the first quadrant bounded by y = x2>3 and y = 4 34. The region in the first quadrant bounded by y = 2 and

y = 2 sin x on the interval 30, p>24 35. The region bounded by y = ex, y = 2e-x + 1, and x = 0

36. The region below the line y = 2 and above the curve y = sec2 x on the interval 30, p>44

37. The region between the line y = x and the curve y = 2×21 – x2 in the first quadrant

38. The region bounded by x = y2 – 4 and y = x>3 Further Explorations 39. Explain why or why not Determine whether the following state-

ments are true and give an explanation or counterexample.

a. The area of the region bounded by y = x and x = y2 can be found only by integrating with respect to x.

b. The area of the region between y = sin x and y = cos x on the interval 30, p>24 is 1p>20 1cos x – sin x2 dx.

c. 110 1x – x22 dx = 110 11y – y2 dy.

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6.2 Regions Between Curves 419

65. Lorenz curves and the Gini index A Lorenz curve is given by y = L1x2, where 0 … x … 1 represents the lowest fraction of the population of a society in terms of wealth and 0 … y … 1 repre- sents the fraction of the total wealth that is owned by that fraction of the society. For example, the Lorenz curve in the figure shows that L10.52 = 0.2, which means that the lowest 0.5 (50%) of the society owns 0.2 (20%) of the wealth. (See the Guided Project Distribution of Wealth for more on Lorenz curves.)

Line of perfect equality

0 1.00.80.5

1.0

0.6

0.2

y

x

Fraction of households

A

B

Fr ac

tio n

of to

ta l i

nc om

e

0.8 of households control 0.6 of the wealth

Lorenz curve, y ! L(x)

0.5 of households control 0.2 of the wealth

a. A Lorenz curve y = L1x2 is accompanied by the line y = x, called the line of perfect equality. Explain why this line is given this name.

b. Explain why a Lorenz curve satisfies the conditions L102 = 0, L112 = 1, L1x2 … x, and L′1×2 Ú 0 on 30, 14.

c. Graph the Lorenz curves L1x2 = xp corresponding to p = 1.1, 1.5, 2, 3, 4. Which value of p corresponds to the most equitable distribution of wealth (closest to the line of perfect equality)? Which value of p corresponds to the least equitable distribution of wealth? Explain.

d. The information in the Lorenz curve is often summarized in a single measure called the Gini index, which is defined as follows. Let A be the area of the region between y = x and y = L1x2 (see figure) and let B be the area of the region between y = L1x2 and the x-axis. Then the Gini index is G =

A A + B. Show that G = 2A = 1 – 2L

1

0

L1x2 dx. e. Compute the Gini index for the cases L1x2 = xp and

p = 1.1, 1.5, 2, 3, 4. f. What is the smallest interval 3a, b4 on which values of the

Gini index lie for L1x2 = xp with p Ú 1? Which endpoints of 3a, b4 correspond to the least and most equitable distribution of wealth?

g. Consider the Lorenz curve described by L1x2 = 5×2>6 + x>6. Show that it satisfies the conditions L102 = 0, L112 = 1, and L′1×2 Ú 0 on 30, 14. Find the Gini index for this function.

Additional Exercises 66. Equal area properties for parabolas Consider the parabola

y = x2. Let P, Q, and R be points on the parabola with R between P and Q on the curve. Let /P, /Q, and /R be the lines tangent to the parabola at P, Q, and R, respectively (see figure). Let P′ be the intersection point of /Q and /R, let Q′ be the intersection point of /P and /R, and let R′ be the intersection point of /P and /Q. Prove

T54.

x

y

O

y ! x2

x ! 2 sin2 y

55.

x

y

O

x ! (y ” 2)2

3

y ! 8 ” x

56–59. Roots and powers Find the area of the following regions, expressing your results in terms of the positive integer n Ú 2.

56. The region bounded by f 1×2 = x and g1x2 = xn, for x Ú 0 57. The region bounded by f 1×2 = x and g1x2 = x1>n, for x Ú 0 58. The region bounded by f 1×2 = x1>n and g1x2 = xn, for x Ú 0 59. Let An be the area of the region bounded by f 1×2 = x1>n and

g1x2 = xn on the interval 30, 14, where n is a positive integer. Evaluate lim

nS∞ An and interpret the result.

60–63. Bisecting regions For each region R, find the horizontal line y = k that divides R into two subregions of equal area.

60. R is the region bounded by y = 1 – x, the x-axis, and the y-axis.

61. R is the region bounded by y = 1 – # x – 1 # and the x-axis.

62. R is the region bounded by y = 4 – x2 and the x-axis.

63. R is the region bounded by y = 1x and y = x. Applications 64. Geometric probability Suppose a dartboard occupies the square 51x, y2: 0 … # x # … 1, 0 … # y # … 16. A dart is thrown randomly

at the board many times (meaning it is equally likely to land at any point in the square). What fraction of the dart throws land closer to the edge of the board than the center? Equivalently, what is the probability that the dart lands closer to the edge of the board than the center? Proceed as follows.

!1

R

0 x1 !1

1

!1

y

0 x1

1

!1

y

R1

C

a. Argue that by symmetry, it is necessary to consider only one quarter of the board, say the region R: 51x, y2: # x # … y … 16.

b. Find the curve C in this region that is equidistant from the cen- ter of the board and the top edge of the board (see figure).

c. The probability that the dart lands closer to the edge of the board than the center is the ratio of the area of the region R1 above C to the area of the entire region R. Compute this probability.

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420 Chapter 6 Applications of Integration

71. Area function for a cubic Consider the cubic polynomial f 1×2 = x1x – a21x – b2, where 0 … a … b. a. For a fixed value of b, find the function F 1a2 = 1b0 f 1×2 dx.

For what value of a (which depends on b) is F 1a2 = 0? b. For a fixed value of b, find the function A1a2 that gives the

area of the region bounded by the graph of f and the x-axis between x = 0 and x = b. Graph this function and show that it has a minimum at a = b>2. What is the maximum value of A1a2, and where does it occur (in terms of b)?

72. Differences of even functions Assume f and g are even, integrable functions on 3-a, a4, where a 7 1. Suppose f 1×2 7 g1x2 7 0 on 3-a, a4 and the area bounded by the graphs of f and g on 3-a, a4 is 10. What is the value of

11a0 x1 f 1×22 – g1x222 dx? 73. Roots and powers Consider the functions f 1×2 = xn and

g1x2 = x1>n, where n Ú 2 is a positive integer. a. Graph f and g for n = 2, 3, and 4, for x Ú 0. b. Give a geometric interpretation of the area function

An1x2 = 1 x0 1 f 1s2 – g1s22 ds, for n = 2, 3, 4, cand x 7 0. c. Find the positive root of An1x2 = 0 in terms of n. Does the

root increase or decrease with n?