you have a function: y=2/x
volume = Integral pi*r^2 dx
where r = y = 2/x
pi*Int(4/x^2 dx) [1,oo]
now take it from there. piece of cake.
Let R denote the region in the plane consisting of all points (x,y) where x ≥ 1 and 0 ≤ y ≤ 2/x. Let S denote the solid formed by rotating R about the x-axis. Observe that S is an unbounded region; that is, it extends indefinitely in the direction of the positive x-axis. Nevertheless, its volume is finite and can be expressed in terms of the improper integral
I just have to set up the integral from 1 to infinity but I don't understand how I can do it without a function
2 answers
Not done is inevitable This post reoanstes with me once I see how something can be done, and prove to myself that I can do it, I too may often abandon it.I've a few thoughts, beyond the obvious focus and finish : enjoy your doodling, abandon earlier, and share on wikis. In detail:Firstly, I would consider this more kindly a proof-of-concept execution is like an artist's sketch, and while it may be a start of a finished work, it may also be a carefully worked doodle: there is certainly fun, and some value, in working something out even if nothing comes of it, as you note. So give yourself a break!Secondly, one issue is that your time is limited and that these half-finished projects are a serious drain. To reduce the time drain, abandon projects earlier! Simply, be satisfied or distracted earlier, without spending such time filling in details that will never be completed to your satisfaction. Cut your losses!Thirdly, may I encourage you to contribute more to wikis? Your (very nice) explanation of the Kelly criterion would make a wonderful (section of) a Wikibook (, say), and one of the reasons wikis work so well is because they combine different spurts of effort some-one else can pick up and build from your abandoned work. Help us build together!More fundamentally, you're noting a conflict between what you find emotionally fulfilling (your id: learning, figuring things out), and what you find rationally fulfilling (your superego: accomplishing, benefiting others). This emotional conflict, not poor task-management, is the root issue, and might be reconciled or managed.Lastly, might I counsel some Japanese philosophy, specifically the aesthetic of ? All that exists is imperfect, impermanent, and incomplete, and this shouldn't be a source of sadness rather one may feel a warm melancholy at the inevitable flaws and loose ends of one's works.Thanks again for your thoughtful and provocative post.