?Lorenz curves and the Gini index A Lorenz curve is given by y = L(x), where \(0 \leq x

Draw the graphs of two functions f and g that are continuous and intersect exactly twice on an interval [a, b]. Explain how to use integration to find the area of the region bounded by the two curves.

Draw the graphs of two continuous functions f and g that are continuous and intersect exactly three times on an interval [a, b]. Explain how to use integration to find the area of the region bounded by the two curves.

Make a sketch to show a case in which the area bounded by two curves is most easily found by integrating with respect to x.

Make a sketch to show a case in which the area bounded by two curves is most easily found by integrating with respect to y.

Finding area Determine the area of the shaded region in the following figures.

Finding area Determine the area of the shaded region in the following figures.

Finding area Determine the area of the shaded region in the following figures.

Finding area Determine the area of the shaded region in the following figures.

Regions between curves Sketch the region and find its area. The region bounded by y = 2(x + 1), y = 3(x + 1), and x = 4

Regions between curves Sketch the region and find its area. The region bounded by y = cos x and y = sin x between \(x=\pi / 4\) and \(x=5 \pi / 4\)

Regions between curves Sketch the region and find its area. The region bounded by \(y=e^{x}\), \(y=e^{-2 x}\), and \(x=\ln 4\)

Regions between curves Sketch the region and find its area. The region bounded by y = x and \(y=x^{2}-2\)

Regions between curves Sketch the region and find its area. The region bounded by \(y=\frac{4}{1+x^{2}}\) and y = 1

Regions between curves Sketch the region and find its area. The region bounded by \(y=64 \sqrt{x}\) and \(y=8 x^{2}\)

Compound regions Sketch the following regions (if a figure is not given) and then find the area. The region bounded by y = sin x, y = cos x, and the x-axis between x = 0 and \(x=\pi / 2\)

Compound regions Sketch the following regions (if a figure is not given) and then find the area. The regions between y = sin x and y = sin 2x, for \(0 \leq x \leq \pi\)

Compound regions Sketch the following regions (if a figure is not given) and then find the area. The region bounded by y = x, y = 1/x, y = 0, and x = 2

Compound regions Sketch the following regions (if a figure is not given) and then find the area. The region in the first quadrant bounded by \(y=(x-1)^{3}\) and y = x - 1

Compound regions Sketch the following regions (if a figure is not given) and then find the area. The region bounded by y = 1 - |x| and the x-axis

Compound regions Sketch the following regions (if a figure is not given) and then find the area. The regions bounded by \(y=x^{3}\) and y = 9x

Compound regions Sketch the following regions (if a figure is not given) and then find the area. The region bounded by y = |x - 3| and y = x/2

Compound regions Sketch the following regions (if a figure is not given) and then find the area. The regions bounded by \(y=x^{2}(3-x)\) and y = 12 - 4x

Integrating with respect to y Sketch the following regions (if a figure is not given) and find the area. The region bounded by y = 8 - 2x, y = x + 8 , and y = 0

Integrating with respect to y Sketch the following regions (if a figure is not given) and find the area. The region bounded by y = ln x, y = 0, and x = 0

Integrating with respect to y Sketch the following regions (if a figure is not given) and find the area. The region bounded by \(y=4-x^{2}\) and y - x - 2 = 0

Integrating with respect to y Sketch the following regions (if a figure is not given) and find the area. The region bounded by \(y=\ln x^{2}\), y = ln x, and \(x=e^{2}\)

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.

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.

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.

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.

Two approaches Find the area of the following regions by (a) integrating with respect to x, and then (b) integrating with respect to y. Be sure your results agree. In each case, sketch the bounding curves and the region in question. The region bounded by \(y=\sqrt{x}\), y = 2x - 15, and y = 0

Two approaches Find the area of the following regions by (a) integrating with respect to x, and then (b) integrating with respect to y. Be sure your results agree. In each case, sketch the bounding curves and the region in question. The region in the first quadrant bounded by \(y=x^{1 / 3}\), 18y - x - 27 = 0, and y = 0

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. The region in the first quadrant bounded by \(y=x^{2 / 3}\) and y = 4

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. The region in the first quadrant bounded by y = 2 and y = 2 sin x on the interval \([0, \pi / 2]\)

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. The region bounded by \(y=e^{x}\), \(y=2 e^{-x}+1\), and x = 0

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. The region below the line y = 2 and above the curve \(y=\sec ^{2} x\) on the interval \([0, \pi / 4]\)

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. The region between the line y = x and the curve \(y=2 x \sqrt{1-x^{2}}\) in the first quadrant

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. The region bounded by \(x=y^{2}-4\), y = x/3.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The area of the region bounded by y = x and \(x=y^{2}\) 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 \([0, \pi / 2]\) is \(\int_{0}^{\pi / 2}(\cos x-\sin x) d x\). c. Without evaluating integrals, \(\int_{0}^{1}\left(x-x^{2}\right) d x=\int_{0}^{1}(\sqrt{y}-y) d y\)

Regions between curves Sketch the region and find its area. The region bounded by y = sin x and \(y=x(x-\pi)\) for \(0 \leq x \leq \pi\)

Regions between curves Sketch the region and find its area. The region bounded by \(y=(x-1)^{2}\) and y = 7x - 19

Regions between curves Sketch the region and find its area. The region bounded by y = 5/4 and \(y=\frac{1}{\sqrt{1-x^{2}}}\)

Regions between curves Sketch the region and find its area. The region bounded by \(y=x^{2}-2 x+1\) and y = 5x - 9

Either method Use the most efficient strategy for computing the area of the following regions. The region bounded by x = y(y - 1) and x = -y(y - 1)

Either method Use the most efficient strategy for computing the area of the following regions. The region bounded by x = y(y - 1) and y = x/3

Either method Use the most efficient strategy for computing the area of the following regions. The region bounded by \(y=x^{3}\), \(y=-x^{3}\), and 3y - 7x - 10 = 0

Either method Use the most efficient strategy for computing the area of the following regions. The region bounded by \(y=\sqrt{x}\), y = 2x - 15, and y = 0

Either method Use the most efficient strategy for computing the area of the following regions. The region bounded by \(y=x^{2}-4\), 4y - 5x - 5 = 0, and y = 0

Either method Use the most efficient strategy for computing the area of the following regions. The region in the first quadrant bounded by \(y=\frac{5}{2}-\frac{1}{x}\) and y = x.

Either method Use the most efficient strategy for computing the area of the following regions. The region in the first quadrant bounded by \(y=x^{-1}\), y = 4x, and y = x/4

Comparing areas Let \(f(x)=x^{p}\) and \(g(x)=x^{1 / q}\), where p > 1 and q > 1 are positive integers. Let \(R_{1}\) be the region in the first quadrant between y = f(x) and y = x and let \(R_{2}\) be the region in the first quadrant between y = g(x) and y = x. a. Find the area of \(R_{1}\) and \(R_{2}\) when p = q, and determine which region has the greater area. b. Find the area of \(R_{1}\) and \(R_{2}\) when p > q, and determine which region has the greater area. c. Find the area of \(R_{1}\) and \(R_{2}\) when p < q, and determine which region has the greater area.

Complicated regions Find the area of the regions shown in the following figures.

Complicated regions Find the area of the regions shown in the following figures.

Complicated regions Find the area of the regions shown in the following figures.

Complicated regions Find the area of the regions shown in the following figures.

Roots and powers Find the area of the following regions, expressing your results in terms of the positive integer \(n \geq 2\). The region bounded by f(x) = x and \(g(x)=x^{n}\), for \(x \geq 0\)

Roots and powers Find the area of the following regions, expressing your results in terms of the positive integer \(n \geq 2\). The region bounded by f(x) = x and \(g(x)=x^{1 / n}\), for \(x \geq 0\)

Roots and powers Find the area of the following regions, expressing your results in terms of the positive integer \(n \geq 2\). The region bounded by \(f(x)=x^{1 / n}\) and \(g(x)=x^{n}\), for \(x \geq 0\)

Roots and powers Find the area of the following regions, expressing your results in terms of the positive integer \(n \geq 2\). Let \(A_{n}\) be the area of the region bounded by \(f(x)=x^{1 / n}\) and \(g(x)=x^{n}\) on the interval [0, 1], where n is a positive integer. Evaluate \(\lim _{n \rightarrow \infty} A_{n}\) and interpret the result.

Geometric probability Suppose a dartboard occupies the square \(\{(x, y): 0 \leq|x| \leq 1,0 \leq|y| \leq 1\}\). A dart is thrown randomly at the board many times (meaning it is equally likely to land at any point in the square). In what fraction of the throws does the dart 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. a. Argue that by symmetry it is necessary to consider only one quarter of the board, say the region \(R:\{(x, y):|x| \leq y \leq 1\}\). b. Find the curve C in this region that is equidistant from the center 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 \(R_{1}\) above C to the area of the entire region R. Compute this probability.

Lorenz curves and the Gini index A Lorenz curve is given by y = L(x), where \(0 \leq x \leq 1\) represents the lowest fraction of the population of a society in terms of wealth and \(0 \leq y \leq 1\) represents 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 L(0.5) = 0.2, which means that the lowest 0.5(50%) of the society owns 0.2(20%) of the wealth. (See Guided Projects for more on Lorenz curves.) a. A Lorenz curve y = L(x) 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 L(0) = 0, L(1) = 1, and \(L^{\prime}(x) \geq 0\) on [0, 1]. c. Graph the Lorenz curves \(L(x)=x^{p}\) 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 = L(x) (see figure) and let B be the area of the region between y = L(x) and the x-axis. Then the Gini index is \(G=\frac{A}{A+B}\). Show that \(G=2 A=1-2 \int_{0}^{1} L(x) d x\). e. Compute the Gini index for the cases \(L(x)=x^{p}\) and p = 1.1, 1.5, 2, 3, 4. f. What is the smallest interval [a, b] on which values of the Gini index lie, for \(L(x)=x^{p}\) with \(p \geq 1\)? Which endpoints of [a, b] correspond to the least and most equitable distribution of wealth? g. Consider the Lorenz curve described by \(L(x)=5 x^{2} / 6+x / 6\). Show that it satisfies the conditions L(0) = 0, L(1) = 1, and \(L^{\prime}(x) \geq 0\) on [0, 1]. Find the Gini index for this function.

Equal area properties for parabolas Consider the parabola \(y=x^{2}\). Let P, Q, and R be points on the parabola with R between P and Q on the curve. Let \(\ell_{P}\), \(\ell_{Q}\), and \(\ell_{R}\) be the lines tangent to the parabola at P, Q, and R, respectively (see figure). Let \(P^{\prime}\) be the intersection point of \(\ell_{Q}\), and \(\ell_{R}\); let \(Q^{\prime}\) be the intersection point of \(\ell_{P}\) and \(\ell_{R}\); and let \(R^{\prime}\) be the intersection point of \(\ell_{P}\) and \(\ell_{Q}\). Prove that Area \(\Delta P Q R=2 \cdot \text{ Area }~\Delta P^{\prime} Q^{\prime} R^{\prime}\) in the following cases. (In fact, the property holds for any three points on any parabola.) (Mathematics Magazine 81, No. 2 (April 2008): 83-95.) a. \(P\left(-a, a^{2}\right)\), \(Q\left(a, a^{2}\right)\), and R(0, 0), where a is a positive real number b. \(P\left(-a, a^{2}\right)\), \(Q\left(b, b^{2}\right)\), and R(0, 0), where a and b are positive real numbers c. \(P\left(-a, a^{2}\right)\), \(Q\left(b, b^{2}\right)\), and R is any point between P and Q on the curve

Minimum area Graph the curves y = (x + 1)(x - 2) and y = ax + 1 for various values of a. For what value of a is the area of the region between the two curves a minimum?

An area function Graph the curves \(y=a^{2} x^{3}\) and \(y=\sqrt{x}\) for various values of a > 0. Note how the area A(a) between the curves varies with a. Find and graph the area function A(a). For what value of a is A(a) = 16?

Area of a curve defined implicitly Determine the area of the shaded region bounded by the curve \(x^{2}=y^{4}\left(1-y^{3}\right)\) (see figure).

Rewrite first Find the area of the region bounded by the curve \(x=\frac{1}{2 y}-\sqrt{\frac{1}{4 y^{2}}-1}\) and the line x = 1 in the first quadrant. (Hint: Express y in terms of x.)

Area function for a cubic Consider the cubic polynomial f(x) = x(x-a)(x-b), where \(0 \leq a \leq b\). a. For a fixed value of b, find the function \(F(a)=\int_{0}^{b} f(x) d x\). For what value of a (which depends on b) is F(a) = 0? b. For a fixed value of b, find the function A(a) 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 A(a), and where does it occur (in terms of b) ?

Differences of even functions Assume f and g are even, integrable functions on [-a, a], where a > 1. Suppose f(x) > g(x) > 0 on [-a, a] and that the area bounded by the graphs of f and g on [-a, a] is 10 . What is the value of \(\int_{0}^{\sqrt{a}} x\left[f\left(x^{2}\right)-g\left(x^{2}\right)\right] d x\)?

Roots and powers Consider the functions \(f(x)=x^{n}\) and \(g(x)=x^{1 / n}\), where \(n \geq 2\) is a positive integer. a. Graph f and g for n = 2, 3, and 4 for \(x \geq 0\). b. Give a geometric interpretation of the area function \(A_{n}(x)=\int_{0}^{x}(f(s)-g(s)) d s\) for n = 2, 3, 4, . . . and x > 0. c. Find the positive root of \(A_{n}(x)=0\) in terms of n. Does the root increase or decrease with n?

Shifting sines Consider the functions f(x) = a sin 2x and g(x) = (sin x)/a, where a > 0 is a real number. a. Graph the two functions on the interval \([0, \pi / 2]\) for \(a=\frac{1}{2}\), 1 , and 2. b. Show that the curves have an intersection point \(x^{*}\) (other than x = 0) on \([0, \pi / 2]\) that satisfies \(\cos x^{*}=1 /\left(2 a^{2}\right)\), provided \(a \geq 1 / \sqrt{2}\). c. Find the area of the region between the two curves on \(\left[0, x^{*}\right]\) when a = 1. d. Show that as \(a \rightarrow 1 / \sqrt{2}\), the area of the region between the two curves on \(\left[0, x^{*}\right]\) approaches zero.