Here's one method:
Each perpendicular slice to the z-axis for |z|<c is an ellipse whose equation is
The area of which is:
Find with proof the volume of the region that is contained within and with .
Sure I know this can be done with calculus (multiple integrals or volume from rotation) but if that were the kind of solution I was looking for I would have posted in the Calculus section.
Hint: Orthogonal Projections
If you want I can show you how these approximation arguments lead to a problem.
I have a different problem. What is the perimeter of an ellipse?
ok here is how it is done without calculus.
before i start i must first find the area of the largest triangle that will fit into an ellipse with major axis of length 2a and minor axis of length 2b also note that the formula for the area of an ellipse is where 2a and 2b are the length of the axies. using orthogonal projectiles we make the ellipse into a circle with radius of length b. since this kind of projection preserves ratios of areas we have
(ABC) is the area of the largest triangle that can fit into the circle, this area is (i assume that you can prove this on your own). (xyz) is the area of the larges triangle that can fit into the ellipse. so solving for (xyz) we have
now for the original problem.
WLOG assume that
once again we make a projection that converts the ellipsoid into a sphere of radius a. now consider the area of the largest tetrahedron that fits into the upper half of the sphere. the area of this is (ABCD).
now think about (wxyz) the area of the larges tetrahedron that will fit into the upper half of the ellipse, with base on the face where c=0. it is plain to see that the volume of this is expressed as
now using the fact the ratio of volumes are preserved we have
solving for we have
and V is the volume of the ellipsoid.
like i said in the hint, use orthogonal projections, and calculus was not needed.