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Volumes of Revolution
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Finding the volume of a solid revolved around a function or axis to create a 3D figure
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The volume produced by a 3D shape which is symmetrical around the x-axis can be approximated by slicing it up into tiny cylinders, and adding them up.
We can make this approximation exact by making the cylinder infinitely small and integrating.
Volume of revolution about x-axis
AHL 5.16
A curve y=f(x) can be revolved around the x-axis to produce a 3D solid. The following example shows y=2+sinx revolved 2π about the x-axis.
The volume of the resulting solid is given by
V=∫abπy2dx📖
Volume of revolution about y-axis
AHL 5.16
The volume of the solid produced by revolving a curve 2π about the y-axis by finding x in terms of y and evaluating
V=∫abπx2dy📖
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