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March 29, 2024, 05:02:29 pm

Author Topic: Volume of a Hemisphere using Parallel Cross-Sections of a similar shape  (Read 1107 times)  Share 

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frog0101

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Hi,
how would you do this question by Parallel Cross-Sections (I know its way easier to do by cylindrical shells and by slices but just curious)?
Find the volume of a hemisphere of radius a units.

jakesilove

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Hi,
how would you do this question by Parallel Cross-Sections (I know its way easier to do by cylindrical shells and by slices but just curious)?
Find the volume of a hemisphere of radius a units.

For simplicity, I'll find the volume of a sphere (the volume of a hemisphere will just be half of this). See the below diagram.



Each slice has area

and we can show (by Pythag) that



Therefore, the cross section as a function of x will be


Then, we do our usual dx limit thingy, which gets us to

Evaluate this integral, and you get

As expected!





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RuiAce

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Small remark: When it comes to dealing with the hemisphere or the sphere, the parallel cross sections method works exactly the same way as the discs/washers method.

Why this happens
It has to do with how the cross sections are circular.

Cross sections that are circular form the basis of where the volumes by discs appear. This is because we have what's known as a "volume of solid of revolution"; the type of volume that's first presented in 2U.

Taking a parallel cross section will be equivalent to applying the discs method only when the cross sections we seek are circular.

Depending on how you choose your orientations, the two methods will produce the same integral.