MHB Rotation around a curve. Find the Volume.

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To find the volume generated by rotating a curve around itself, the function must be continuous to ensure meaningful results. The term "rotated around itself" needs clarification, as it can lead to confusion regarding the shape being analyzed. For example, rotating a constant function yields no volume since it forms a line. The discussion emphasizes the importance of understanding the nature of the function and its properties when calculating volume. Overall, the volume of a curve depends significantly on its characteristics and the method of rotation.
cbarker1
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I am thinking about how to find the volume rotate around its function.Let f be a function of x in the interval [a,b] . The function could be any curve. And the curve is rotation around itself. Would there exist a volume of the curve? And how to find the volumeThank you

CBARKER1
 
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Well, here are some thoughts. We need some regularity on the function, such as continuity, otherwise this can be very messy and the question won't make sense. Second, what do you mean by "rotated around itself"? If you consider a simple case, i.e. a constant function, there's no volume. You're just rotating a line.
 
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Thread 'Problem with calculating projections of curl using rotation of contour'

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