Let’s say we need to calculate the volume of an ellipsoid:
This ellipsoid intersects the x, y, z axes at coordinates 2, 
and 3, respectively (the general equation of the ellipsoid is: 
, where a, b, c are the coordinates of the intersections).
This is not a rotational ellipsoid, not formed by rotating any curve around any axis, so we can’t use the standard formula for the volume of a rotational solid:
We need to figure out another way.
1. Choose any point M(z) at the center of the ellipsoid and on the OZ axis.
A plane passing through this point and perpendicular to the OZ axis “cuts” out a certain ellipse from the ellipsoid:
2. Determine the equation of the “cut-out” ellipse projection on the XY plane
We determine the equation of this ellipse, for a fixed ‘z’ (treating ‘z’ as a constant) from the general equation of the ellipsoid:
You can see that our ‘a’ and ‘b’ from the general equation of the ellipsoid (
) are:
4. Calculate the area of this section as a function of ‘z’
The area of this ellipse depends on the chosen point ‘z’, so it will be a function of ‘z’. We can calculate it either using the ready formula for the area of an ellipse (
):
Or by laboriously calculating the appropriate definite integral (using, of course, the parametric form of the ellipse and the formula for the area of a region in parametric form):
5. Calculate the volume of the solid using the cross-sectional areas
Now comes the tricky part. The volume of the solid is equal to – this sounds a bit awkward – the “sum” (i.e., the integral) of all the cross-sections, which in general is:
where 
is the function of the cross-sectional areas of the solid with a plane perpendicular to the OZ axis, and ‘a’ and ‘b’ are the bounds within which ‘z’ varies.
So for us:
= (calculate, calculate, calculate…) = 
This matches the general formula for the ellipsoid (
).
THE END
It’s worth remembering this general scheme, and most importantly, that the volume of more complex, non-rotational solids can be calculated by integrating the function of their cross-sectional areas.
