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What are volumes of revolution and how are they calculated?


Volumes of revolution are contructed by a fuction of x, y=f(x), rotated 360 degrees, or 2pi radians around the x asxis. For example, a function of y=2 would create a cylinder of radius 2 in its volume of revolution, and the function y=x would create a cone in its volume of revolution. This can be seen by visualising roatating to function in and out of the page using the x-axis as a centre. As the x-axis is the centre of rotation, the radius of the volume of revolution at the point will simply be equal to the value of f(x), the cross sectional area at that point will be A=pi*r2, or A=pi*f(x)2. The sum of all these areas between two x-vales should give the total volume of revolution between those two x values, this continuous sum can be defined as:


integral between (x1,x0) of pi*f(x) dx.

For example, considering a cone of radius r and height h

x1=h and  x0=0, f(0)=0 and f(h)=r, therefore f(x)=rx/h

Therefore

volume = integral between (h,0) of pi*r2x2/hdx

=0[pi*r2x3/3h2]h=pi*r2h/3

This is the correct formula for the Volume of a cone, showing the integration method works.

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