Shown in Fig. 1 is a pile of cylinders that approximates one half of a sphere.
Fig. 1
From Fig. 1, we see .
If denotes the volume of
cylinder in the pile, then
.
Let
,
we have
see “Little Bird and a Recursive Generator“
.
As
Therefore, gives
Similarly, we approximate a cone
by a stack of cylinders:
Fig. 2
Since
and
,
we have
If denotes the
cylinder, then
.
Let
we have
.
Since as ,
, we have
We now derive the formula for a sphere’s surface area.
Let us approximate a sphere by many cones:
Fig. 3
From Fig. 3, we see that as ,
.
Consequently,
and,
.
It follows that
Hence,