91Ó°ÊÓ

Find the volume of the solid whose base is bounded by the circle \(x^{2}+y^{2}=4\) with the indicated cross sections taken perpendicular to the \(x\) -axis. (a) Squares (b) Equilateral triangles

The base of a solid is bounded by \(y=x^{3}, y=0\), and \(x=1\). Find the volume of the solid for each of the following cross sections (taken perpendicular to the \(y\) -axis): (a) squares, (b) semicircles, (c) equilateral triangles, and (d) semiellipses whose heights are twice the lengths of their bases.

Suspension Bridge A cable for a suspension bridge has the shape of a parabola with equation \(y=k x^{2}\). Let \(h\) represent the height of the cable from its lowest point to its highest point and let \(2 w\) represent the total span of the bridge (see figure). Show that the length \(C\) of the cable is given by \(C=2 \int_{0}^{w} \sqrt{1+\frac{4 h^{2}}{w^{4}} x^{2}} d x\)

Economists use Lorenz curves to illustrate the distribution of income in a country. A Lorenz curve, \(y=f(x)\), represents the actual income distribution in the country. In this model, \(x\) represents percents of families in the country and \(y\) represents percents of total income. The model \(y=x\) represents a country in which each family has the same income. The area between these two models, where \(0 \leq x \leq 100\), indicates a country's "income inequality." The table lists percents of income \(y\) for selected percents of families \(x\) in a country. $$ \begin{aligned} &\begin{array}{|c|c|c|c|c|c|} \hline x & 10 & 20 & 30 & 40 & 50 \\ \hline y & 3.35 & 6.07 & 9.17 & 13.39 & 19.45 \\ \hline \end{array}\\\ &\begin{array}{|c|c|c|c|c|} \hline x & 60 & 70 & 80 & 90 \\ \hline y & 28.03 & 39.77 & 55.28 & 75.12 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility to find a quadratic model for the Lorenz curve. (b) Plot the data and graph the model. (c) Graph the model \(y=x .\) How does this model compare with the model in part (a)? (d) Use the integration capabilities of a graphing utility to approximate the "income inequality."