Approximating the Volume of a Sphere using Cylinders
The volume of a sphere can be approximated by cylinders. As the number of cylinders approaches infinity the total volume of all the cylinder approaches the actual volume of the sphere.
Using integration, the exact volume of the sphere can be found. The exact volume of a sphere is 43⋅π⋅ r3, where r is the radius.
Assume the center of the sphere is located at the origin.
Let r be the radius of the sphere, and let n be the number of approximating cylinders.
The height h of each cylinder can be defined as:
h = 2⋅rn
The radius rk of the kth cylinder located at height yk can be found from
The volume of the kth cylinder is thus:
Vk = π⋅rk2⋅h = π⋅r2−yk2⋅h
Therefore the total volume of n cylinders is:
Volume = ∑k=1nπ ⋅r2−yk2⋅h
In the limit as n → ∞, we get the volume of the sphere:
Vsphere = ∫−rrπ ⋅r2−y2 dy
= 43⋅π⋅ r3
Change the number of cylinders used to approximate the volume of the sphere:
Number of Cylinders =
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