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Method of disks or the method of rings
One of the simple methods for getting the cross-sectional area is to cut the object perpendicular to the axis of rotation. Carrying out this the cross section will be either a solid disk if the object is solid (as our above example is) or a ring if we've hollowed out portion of the solid (we will illustrates this eventually).
In the case that we obtain a solid disk the area is,
A = ∏ ( radius )2
where the radius will based upon the function and the axis of rotation.
In the case that we get a ring the area is following,
where again both of the radii will based on the functions given & the axis of rotation. Note that in the case of solid disk we can think of the inner radius as zero & we'll arrive at the correct formula for solid disk and therefore this is a much more general formula to utilize.
Also, in both of the cases, whether the area is a function of x or a function of y will based upon the axis of rotation as we will illustrates.
This method is frequently called the method of disks or the method of rings.
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