Renegade343 Posted October 10, 2015 Share Posted October 10, 2015 (edited) For anyone studying calculus, he/she would have learned about the volume of revolution for a function, which is given by the formula: V = ∫πy2 dx Now, half a year ago, I made an extension of the volume of revolution formula in this thread. Now, with more obtained math knowledge, I have managed to generalise the formula to be useful for all practical purposes. So, with the formula above, it assumes that we will be adding lots of circular disks with a depth of dx. However, what if we have this? What happens when we want to find the volume bounded by the function when it is rotated around the x-axis, and not making circular disks? By intuition, we would expect that the volume of revolution formula for adding lots of elliptical disks to be: V = ∫πab dx, a = major axis, b = minor axis (can be interchangeable) Of course, in this hypothetical lesson, two questions are thrown my way: 1. How do we find the area of the ellipse? 2. How do we find the formula of the ellipse? We shall answer question 1 below: And then, question 2 (with a diagram): Now that those two are cleared up, we should notice something. Like circular disks expanding in radius proportional to the function, we also see this happening with elliptical disks: From here, we see that when we move from x = 1 to x = 1.5, the minor axis (parallel to the z-axis) increases by 0.5, and since the major axis (parallel to the y-axis) is twice as long as the minor axis, when we move from x = 1 to x = 1.5, it increases by 1. This follows the explanation of how the original volume of revolution formula came to be. From that example, we can divide the minor radius by the major radius to obtain a constant, in which we can do this (typo: "parallel to y-axis" should be "parallel to z-axis"): This concludes Part 1 of the generalisation of the volume of revolution formula. Edited October 10, 2015 by Renegade343 Link to comment Share on other sites More sharing options...
OverlordMcGeek Posted October 10, 2015 Share Posted October 10, 2015 I honestly didn't think you would keep your word. Nicely done. Link to comment Share on other sites More sharing options...
Insizer Posted October 10, 2015 Share Posted October 10, 2015 step 3: make a Matlab program to analyze pics and do the calculations for you. Link to comment Share on other sites More sharing options...
Renegade343 Posted October 10, 2015 Author Share Posted October 10, 2015 step 3: make a Matlab program to analyze pics and do the calculations for you. You need to know why that happens though. Besides, not everyone can have MATLAB on the go. Link to comment Share on other sites More sharing options...
FelisImpurrator Posted October 10, 2015 Share Posted October 10, 2015 I honestly didn't think you would keep your word. Nicely done. I REALLY want to know the context of this. Link to comment Share on other sites More sharing options...
Eftycchia Posted October 10, 2015 Share Posted October 10, 2015 Unknown context, but the entire OP is solidly built and easy to understand. Main gripe with the thread being images not spoiler'd. Waiting for Part 2. Link to comment Share on other sites More sharing options...
Renegade343 Posted October 10, 2015 Author Share Posted October 10, 2015 And strangely enough, after doing semi-deep searches around the Internet, I seem to be the only one that found the volume of revolution formula when using elliptical disks. Also strangely enough, after reading a few calculus/general math textbooks, there is also no mention of my formula being made before. I wonder if I made something new, or it was so trivial that everyone assumed anyone can find the relationship. Link to comment Share on other sites More sharing options...
FelisImpurrator Posted October 10, 2015 Share Posted October 10, 2015 And strangely enough, after doing semi-deep searches around the Internet, I seem to be the only one that found the volume of revolution formula when using elliptical disks. Also strangely enough, after reading a few calculus/general math textbooks, there is also no mention of my formula being made before. I wonder if I made something new, or it was so trivial that everyone assumed anyone can find the relationship. A) Submit that S#&$ somewhere. Somewhere Most Official. You might win a math award out of nowhere. Or be told that this doesn't need a formula... uh, whatever it is. (I am not a math person.) But either way, at least you'll know. B) Why is this just now a thing? What is the context behind this thread? I can't understand any of it, but damned if it isn't interesting. Link to comment Share on other sites More sharing options...
Renegade343 Posted October 10, 2015 Author Share Posted October 10, 2015 A) Submit that S#&$ somewhere. Somewhere Most Official. You might win a math award out of nowhere. Or be told that this doesn't need a formula... uh, whatever it is. (I am not a math person.) But either way, at least you'll know. I'll probably just ask the university I study in. B) Why is this just now a thing? What is the context behind this thread? I can't understand any of it, but damned if it isn't interesting. To celebrate Fermat's biggest blunder, I made a thread that mirrors Fermat's blunder (Fermat's Last Theorem), saying that I have found a beautiful proof, but due to something silly, I will not be writing the proof out. However, given the potential for its usefulness for all practical purposes, I thought it would be better to actually post that thing out. Link to comment Share on other sites More sharing options...
TheBrsrkr Posted October 10, 2015 Share Posted October 10, 2015 Math: Not even once Link to comment Share on other sites More sharing options...
Eftycchia Posted October 11, 2015 Share Posted October 11, 2015 Math: Not even once It just works. Link to comment Share on other sites More sharing options...
Recommended Posts