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Geometrically the definite integral from a to b is the area under the curve and the double integral is the volume under the surface. So just taking the integral of a function does not yield the volume of the solid made by rotating it around an axis. An integral is only a solid of revolution if you take an infinite sum of infinitesimally small cylinders that is the disk method or you do the same with shells.

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Q: Is integration a way of calculating the volume of the same function equation with an additional dimension or is it always completely new volume rotated about an axis ie a circle to a sphere?
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