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