Rotation around a curve. Find the Volume.

In summary, the conversation discusses finding the volume of a curve that is rotated around itself. It is mentioned that the function must have some regularity, such as continuity, for the question to make sense. It is also noted that in some cases, such as a constant function, there is no volume as it is simply rotating a line.
  • #1
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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  • #2
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.
 

FAQ: Rotation around a curve. Find the Volume.

How do you calculate the volume of a curved shape?

The volume of a curved shape can be calculated by using the formula V = ∫ A(x) dx, where A(x) is the area of the cross-section at a specific point along the curve and dx represents a small change in the x-direction.

What is the difference between rotation around a curve and rotation around a line?

Rotation around a curve involves rotating a shape around a specific curve, while rotation around a line involves rotating a shape around a straight line. The resulting volumes will be different, as the curved shape will have varying cross-sections along the curve, while the straight line will have consistent cross-sections.

Can you rotate a shape around a curve that is not a perfect circle?

Yes, you can rotate a shape around any type of curve as long as you have the equation of the curve and can calculate the corresponding cross-sectional areas.

How do you find the volume of a shape that is rotated around a curve in three dimensions?

To find the volume of a shape that is rotated around a curve in three dimensions, you will need to use a triple integral. This involves integrating the cross-sectional areas in the x, y, and z directions.

What are some real-world applications of calculating volume by rotation around a curve?

Some real-world applications of calculating volume by rotation around a curve include determining the volume of pipes, barrels, and other curved containers, as well as calculating the volume of irregularly shaped objects such as tree trunks and bones.

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