Assume is a surface on
, the domain
is the projection of the surface
on
plane. Then the area of the surface is
where and
are partial derivatives of
with respect to the variable
and
respectively.
If a surface is and the projection of it on
plane is
, then the volume bounded by
plane and the surface
is
Theorem 1.
The surface area of the sphere with radius is
The volume of the sphere with radius is
Proof. The equation of the sphere with radius is
First we calculate the surface area of sphere.
Assume the function
Then
Therefore
The surface area of half-sphere is
Hence, the total surface area of the sphere with radius is
Second, we calculate the volume of the sphere with radius
Theorem 2. The volume of the ellipsoid is
Proof.
The upper bound of the volume is
The lower bound of the volume is
Assume the volume of the ellipsoid is
where we use the substitution and
the determinant of Jacobian matrix is
Therefore, the value equals to