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Approximating the Volume of a Sphere using Cylinders
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Change the number of cylinders used to approximate the volume of the sphere
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Main Concept
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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  , where r is the radius.
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Proof
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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:
The radius ![r[k]](/support/helpjp/helpview.aspx?si=8363/file02224/math93.png) of the kth cylinder located at height ![y[k]](/support/helpjp/helpview.aspx?si=8363/file02224/math97.png) can be found from
![r[k]^2 = r^2-y[k]^2](/support/helpjp/helpview.aspx?si=8363/file02224/math103.png)
The volume of the kth cylinder is thus:
![V[k] = Pi*r[k]^2*h and Pi*r[k]^2*h = Pi*(r^2-y[k]^2)*h](/support/helpjp/helpview.aspx?si=8363/file02224/math120.png)
Therefore the total volume of n cylinders is:
![Volume = sum(Pi*(r^2-y[k]^2)*h, k = 1 .. n)](/support/helpjp/helpview.aspx?si=8363/file02224/math137.png)
In the limit as  , we get the volume of the sphere:
![V[sphere] = (∫)[-r]^(r)Pi *(r^(2)-y^(2)) dy](/support/helpjp/helpview.aspx?si=8363/file02224/math154.png)
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