Find the volume v of the described solid s. the base of s is an elliptical region with boundary curve 16x2 + 9y2 = 144. cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.

In the - plane, the base has equation(s)

which is to say, the distance (parallel to the -axis) between the top and the bottom of the ellipse is

so that at any given , the cross-section has a hypotenuse whose length is .

The cross-section is an isosceles right triangle, which means the legs occur with the hypotenuse in a ratio of 1 to , so that the legs have length . Then the area of each cross-section is

Then the volume of this solid is