91Ó°ÊÓ

Finding Volumes of Solids Find the volumes of the solids (see figures) generated if the upper half of the ellipse \(9 x^{2}+25 y^{2}=225\) is revolved about (a) the \(x\) -axis to form a prolate spheroid (shaped like a football) and (b) the \(y\) -axis to form an oblate spheroid (shaped like half of a candy).

Astroid Find the area of the surface formed by revolving the portion in the first quadrant of the graph of \(x^{2 / 3}+y^{2 / 3}=4\) , \(0 \leq y \leq 8,\) about the \(y\) -axis.

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 \(C=2 \int_{0}^{w} \sqrt{1+\left(\frac{4 h^{2}}{w^{2}}\right) x^{2}} d x\)

Lorenz Curve 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 from the poorest to the wealthiest and \(y\) represents 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. $$\begin{array} { | c | c | c | c | c | c | } \hline x & { 10 } & { 20 } & { 30 } & { 40 } & { 50 } \\ \hline y & { 3.35 } & { 6.07 } & { 9.17 } & { 13.39 } & { 19.45 } \\ \hline \end{array}$$ $$\begin{array} { | c | c | c | c | c | } \hline x & { 60 } & { 70 } & { 80 } & { 90 } \\ \hline y & { 28.03 } & { 39.77 } & { 55.28 } & { 75.12 } \\\ \hline \end{array}$$ (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.

True or False? In Exercises \(83 - 86\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \begin{array} { l } { \text { The line } } \\ { y = ( 1 - \sqrt [ 3 ] { 0.5 } ) x } \\ { \text { divides the region under the curve } } \\ { f ( x ) = x ( 1 - x ) } \\ { \text { on } [ 0,1 ] \text { into two regions of equal area. } } \end{array} $$