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Actually, this isn't too far off. See http://mathworld.wolfram.com/PappussCentroidTheorem.html, where they talk about the 2nd Theorem of Pappus. If you determine the centroid of the region that's being revolved, and then determine the distance the centroid moves in rotation, the volume of the solid of revolution is the product of the distance the centroid moves, and the area of the region being revolved.Terrell said:why can't it be like this?
AHHH! that's it. wedge shape like cutting cakes. i now see why my formula can't work haha! yes it has to vary from the axis of symmetry! thank you for adding more insight into it :DSimon Bridge said:Your intuition comes from the human faculty for pattern recognition. It works well as long as you understand where the patterns come from.
You got the formula you did by following the general pattern ... but the setup was generally fine: you can rotate a function like that, so long at the volume integrated is wedge shaped slices - like cutting a cake. It comes in handy if the shape being cut varies in distance from the symmetry axis ... i.e. what if the cylinder in the example above had an oval hole in it instead of a circular one?
If you explore this area you'll run ahead of your course ;)
i've always wondered if this was how it's supposed to be in geometry class. thank you for bringing that theorem up sir!Mark44 said:Actually, this isn't too far off. See http://mathworld.wolfram.com/PappussCentroidTheorem.html, where they talk about the 2nd Theorem of Pappus. If you determine the centroid of the region that's being revolved, and then determine the distance the centroid moves in rotation, the volume of the solid of revolution is the product of the distance the centroid moves, and the area of the region being revolved.
The concept of finding the volume under the curve of a rotated function is based on the integration of the function over a specified interval and then rotating the resulting curve around a given axis. The volume under the curve can be thought of as the space enclosed by the rotated curve and the axis of rotation.
The volume under the curve of a rotated function is calculated using the method of cylindrical shells. This involves breaking up the function into thin cylindrical shells, calculating the volume of each shell, and then summing up the volumes of all the shells to get the total volume under the curve.
Finding the volume under the curve of a rotated function is important in various fields of science and engineering, such as physics, calculus, and geometry. It allows us to calculate the volume of complex shapes and surfaces, which has practical applications in designing structures and solving real-world problems.
No, the volume under the curve of a rotated function cannot be negative. This is because the volume is a measure of the space enclosed by the rotated curve and the axis of rotation, and space cannot have a negative value.
Yes, there are some limitations to finding the volume under the curve of a rotated function. This method can only be used for functions that are continuous and have a finite number of discontinuities. Additionally, the function should also have a bounded domain over the specified interval for the calculation to be accurate.