# Find the volume V of the described solid S. The base of S is an elliptical region with boundary curve 9x^2 + 16y^2 = 144. Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.

$Because\;the\;cross\;sections\;are\;isosceles\;right\;triangles\;with\\* hypotenuse\;in\;the\;base,\;the\;length\;of\;the\;hypotenuse\;is\;6\sqrt{1-\frac{x^2}{4}},\\*which\;means\;that\;the\;area\;of\;the\;cross\;section\;is\; 9(1-\frac{x^2}{4})\\*\Rightarrow \int_{-2}^{2}9(1-\frac{x^2}{4})\;dx\\* \Rightarrow 9x-\frac{9x^3}{12}= 9(2)-\frac{3\times (2)^3}{4}-9(-2)+ \frac{3\times (-2)^3}{4}\\*\Rightarrow 18-6+18-6=24\\*\therefore required\; area\;will\;be\;24\;sq.\;units$

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