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

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