In the above construction the vertical axis of the disk is divided into m equal intervals. The width of a rectangle is the width of the disk at that height. As the subdivision of the vertical axis of the disk becomes finer and finer the sum of the areas of the rectangles approaches a limit which is called the area of the disk.

The process can be represented algebraically by noting that the equation of the ellipse which encloses the disk can be expressed in terms of the horizontal and vertical distances from the center, x and y, as (x/a)² + (y/b)² = 1. Thus at a height y the width of the disk is 2r where r = a(1-(y/b)²)^1/2. The limit can be expressed as an integral; i.e.,

V₂(a,b) = ∫^b_-b2a(1-(y/b)²)^1/2dy
 

By symmetry this can be reduced to:

V₂(a,b) = 2∫^b₀2a(1-(y/b)²)^1/2dy
 

By a change of variable to z=y/b (and thus dy=bdz) the area reduces to

V₂(a,b) = 2∫¹₀2a(1-z²)^1/2bdz
= 4ab∫¹₀(1-z²)^1/2dz
 

Since ∫(1-z²)^1/2dz = (z/2)(1-z²)^1/2 + (1/2)sin^-1(z) the definite integral reduces to (1/2)π/2 and hence the area of the elliptical disk reduces to

V₂(a,b) = 4ab[(1/2)π/2] = πab
 

For the volume enclosed within an ellipsoid of semi-radii a,b and c the procedure is similar. The equation for the ellipsoid is (x/a)²+(y/b)²+(z/c)²=1. Thus at a level z the cross-section is an elliptical disk enclosed within an ellipse with semi-radii of a(1-(z/c)²)^1/2 and b(1-(z/c)²)^1/2. The volume of the ellipsoid can be expressed as

V₃(a,b,c) = 2∫^c₀V₂(a(1-(z/c)²)^1/2b(1-(z/c)²)^1/2)dz
= 4∫^c₀πab(1-(z/c)²)dz
 

With a change of variable to s=z/c the formula becomes

V₃(a,b,c) = 2πabc∫¹₀(1-s²)ds
= 2πabc = 2πabc[s - (1/3)s³]¹₀
= 2πabc[1-(1/3)] = 2πabc[2/3] = (4/3)πabc
 

For 4 dimensions the formula reduces to

V₄(a,b,c,d) = (8/3)πabcd∫¹₀(1-s²)^3/2ds
 

The definite integral reduces to (3/8)(π/2) so the value of V₄(a,b,c,d) is equal to (1/2)π²abcd.