To obtain the equation for an elliptical region, we divide the boundary curve equation by 36, which gives us:
[9x^2 + 4y^2 = 36] / 36
(x^2)/4 + (y^2)/9 = 1
Since the given cross-sections are isosceles right triangles, the hypotenuse is found on the base. Using trigonometric functions, the hypotenuse is found to be 6 * sqrt(1 - (x^2)/4). The cross-sectional area is then found to be 9 (1 - (x^2)/4).
With the cross-section, we integrate it with limits of -2 to 2 in terms of x to find the volume. This is shown below:
∫ 9 (1 - (x^2)/4) dx (-2,2) = 24
Therefore, the volume is 24 units^3.
