The area of the region bounded by $f(x) = \frac{1}{x}$, $y = 0$, and $x = 1$ is  $$ A = \int_1^{+\infty} f(x) \, \textrm{d}x = \lim_{b \to +\infty} \int_1^b \frac{\textrm{d}x}{x} = \lim_{b \to +\infty} [\ln b - \ln 1] = +\infty $$  The volume of the solid formed by revolving the same region as above is  $$ V = \pi \int_1^{+\infty} [f(x)]^2 \, \textrm{d}x = \lim_{b \to +\infty} \pi \int_1^b \frac{\textrm{d}x}{x^2} = \pi \lim_{b \to +\infty} [-\frac{1}{b} + 1] = \pi $$  I understand that $y = \frac{1}{x}$ and $y = \frac{1}{x^2}$ are different, so it makes perfect algebraic sense that the area under both curves will be different; one being divergent and the other convergent in this question. But how can it make geometric sense that revolving an infinite area results in a finite volume?
Consider the curve $\frac{1}{x}$  where $x \geq 1$. Rotate this curve around the x-axis.   One Dimension - Clearly this structure is infinitely long.  Two Dimensions - Surface Area = $2\pi\int_∞^1\frac{1}{x}dx = 2\pi(\ln ∞ - \ln 1) = ∞$   Three Dimensions -  Volume = $\pi\int_∞^1{x}^{-2}dx = \pi(-\frac{1}{∞} + \frac{1}{1}) = \pi$   So this structure has infinite length and infinite surface area. However it has finite volume, which just does not make sense.  Even more interesting, the "walls" of this structure are infinitely thin. Since the volume is finite, we could fill this structure with a finite amount of paint. To fill the structure the paint would need to cover the complete surface area of the inside of this structure. Since the "walls" are infinitely thin, why would a finite amount of paint not be able to cover the outside of the "walls" too?  Please help me make sense of this whole thing.
Determining bare earth DEM from unclassified LAS file? I have data in LAS format with RGB values created from aerial photogrammetry using a UAV. I am  trying to find a solution to extract the bare earth DEM from the point cloud.   I have tried SAGA, Fusion, MCC-LIDAR, but it is seems they need the LAS file to be already classified (which it naturally isn't). Can anyone point me in the right direction with a brief explanation of the process?   Generally, I would need to process about 100 mill points at a time (can tile them if needed). How do I find the angle between two vectors? I have 3 points on my screen:  a = a point which is (c.x, 0) makes a line pointing straight up b = a user input touch, can be anywhere on the screen c = a moving object         a _______.________ |      |       | |      |       |  |   b  |       | |  .   |       | |   \  |       | |    \ |       |  |     \|       | |      | c     | |______._______|   I have drawn some lines so that you can see the vectors.  I want to be able to get the angle between a and b. I have tried this, but it doesn't work, does anyone know what I'm doing wrong?:  //v1 moving object float boxX = this.mScene.getLastChild().getX();  float boxY = this.mScene.getLastChild().getY();  //v2 user touch float touchX = pSceneTouchEvent.getX(); float touchY = pSceneTouchEvent.getY();       //v3 top of screen float topX = boxX; final float topY = 0;  float dotProd = (touchX * topX) + (touchY * topY);  float sqrtBox = (touchX * touchX) + (touchY * touchY); float sqrtTouch = (topX * topX) + (topY * topY);  double totalSqrt = sqrtBox * sqrtTouch; double theta = Math.acos(dotProd / Math.sqrt(totalSqrt));   The answer I usually get is between 0 and 1. How do I fix this so that I get the angle in degrees? I'm not too sure how to describe the effect that I'm going for or what it is even called.  Thus, I'm having some trouble finding any help through Google.    I want to accomplish something like this "SLOPED" text below, but have the bottoms of the letters follow the slope as well instead of being horizontal.    I have tried using the free transform tool, skewing, and using perspective, but I don't like the effect they are resulting in.  All of the tools that I just mentioned result in the appearance of the word being angled away from the viewer (perspective).  I want more of this style effect, whereas the letters are not skewed horizontally, but scaled evenly along a path that I lay out.  What is the name of this text effect?  Can anyone point me in the right direction to a tutorial or know how to accomplish this themselves?  I know that I could go through and do this letter by letter, but I was really hoping to find an easier method.  EDIT I have removed many elements of my design and quickly threw some sample text in it, but this should give you an idea of what I am trying to accomplish.  I want a mixture of the above and below effects. Prove that the expression cannot be a power of 2 Setting or modifying a (system wide) environment variable in cmd.exe I am looking for a convenient way to add and/or modify and/or delete an environment variable from the command line. Particularly, I find myself at times in situations when I have to add a few variables in cmd.exe.  I'd be grateful if someone showed me a non-GUI way to modify (that is: to add a new directory to) the %PATH% variable.  The change should be be permanent, not just for duration of the cmd session. I just renewed my antivirus (Kaspersky's) yesterday. After that, I logged onto Chemistry.SE (which I frequent often) but then I was greeted with this  Which was definitely not how it looked like earlier that morning. I wasn't able to  Open the reputation and badges tab Vote on questions, answers and comments (even answers to MY questions) Comment on questions or answers. 'Favorite' a question Attach tags to a question Post questions  etc., etc. But when I logged on to the Physics and Biology SEs (which I also frequent), the page format hadn't changed, but I still faced the same problems as above. Does anyone know how to fix this? P.S- I'm not a computer/formatting whiz, so you'd have to try avoiding the jargon when helping me out, sorry.... (Also I use Chrome...in case that's important.) Also @Oded recommended checking out the browser developer tools and having a look at the CSS, but as I said, I'm no Software/formatting whiz...so I'm still lost....  On @Shadow Wizard 's advice I've taken a screen shot of the 'error' notification (I had to scroll down to get all 10 'errors' on screen, hence the need for separate screen-shots) Find Nearby Accounts migration to Google Maps API v3 Why does bash show 'Terminated' after killing a process? Here is the behaviour I want to understand:  $ ps   PID TTY           TIME CMD   392 ttys000    0:00.20 -bash  4268 ttys000    0:00.00 xargs $ kill 4268 $ ps   PID TTY           TIME CMD   392 ttys000    0:00.20 -bash [1]+  Terminated: 15          xargs $ ps   PID TTY           TIME CMD   392 ttys000    0:00.21 -bash   Why does it show the [1]+  Terminated: 15          xargs after I kill a process, instead of just not showing it as it was just killed?  I'm using bash on Mac OS X 10.7.5. To briefly put forward my question, can anyone beautifully explain me in your own view, what was the main intuition behind inventing the ideal of a ring? I want a clarified explanations in these points:   Why is the name "ideal" coined?. In English 'ideal' means "One that is regarded as a standard or model of perfection or excellence." Why did people gave the name of ideal to such group? And why are the ideals not present in the case of groups? And give me a very fantastic intuition and motivation behind the ideals and what are the roles served by them in advanced mathematics.   Thanks a lot, to every one. As it is common with people from my country, I have an immense difficulty with prepositions in English, especially with the use of in and on.   When the preposition indicates the position of the object it is a bit easier to decide which one to use, but as it requires thinking from me, I still say things like "I put the bill in the fridge", "my pencil is on my bag", etc, even though if I think about it, I would know how to say it correctly.  Of course, even in those cases where the preposition indicates position the decision is not always clear, as is the case for example with "on the bus" (you are not on top of the bus, you are inside the bus) or "he is in bed" (you are not inside the bed, you are on top of your bed).  Other cases are even less clear, and I believe there is no real rule to decide what should be said; the person should just know it. Two examples are "in a bad mood", "you are on my mind".  So my first question is whether there is really no rule to help me. The second question refers to two of the cases where I still don't know whether to use in or on:   good luck on/in your exam in/on the list   The last one is especially important to me, as I have asked a number of native speakers, and each time I get a different answer. The only consensus seems to be the case where you say "on top of the list". Apart from that, some people will say that I should say "his name is in the list" whereas others think I should say "his name is on the list". And yeah, a couple of people did tell me that I can use either.  Can someone help me? C, C++ and Java IDE for Ubuntu Is there an IDE which can run on Ubuntu that supports C, C++ and Java? I installed , but it only supported Java. I installed the C/C++ package manually, but that package gives an error if I include iostream.   Is there any other IDE which can satisfy my needs? Or can I get NetBeans with all packages pre-installed in it? How do you show that $e^{-1/x^2}$ is differentiable at $0$? Define the real valued function $f(x)$ by $f(x) = e^{-1/x^2}$ if $x \ne 0$ and $0$ if $x = 0$.    How do you show that $e^{-1/x^2}$ is differentiable at $0$? How do you show that this function is infinitely differentiable? Prove by induction that... $1+3+5+7+...+(2n+1)=(n+1)^2$ for every $n \in \mathbb N$ Is it possible to skip Skyrim's new game introduction cinematic? I don't want to watch the prisoner cart ride to the execution: is it possible to skip this?   If so, how? Can I replace the intro movie file by some empty video? $\sin(n)$ subsequence limits set We were given a challenge by our calculus professor and I've been stuck on it for a while now.   Show that the set of subsequence limits of $A_n=\sin(n)$ is $[-1, 1]$ (another way to phrase this would be: $\forall r\in [-1,1]$ there exists a subsequence of $A_n=\sin(n)$ that converges to $r$).  What would be a good way to start? How to use -exec option in find command I wanna solve some problems in compiz with my brain and hands.  By the way I entered following command to build compiz from source in Ubuntu 12.04  find /opt/compiz-built/share/gconf/schemas -exec gconftool-2 --install-schema-file={};   I referred that command at   And following message is shown.  How to use -exec option in find command, I guess that it's my mistake. What are the differences between "-" and "--" in commands? I often see a single dash - or double dash -- followed by linux commands or other installed linux programs.   What are the differences between single dash and double dashes?  Is it just about abbreviation?  For example, -v means --verbose most of the time. Change default CSS of Google Chrome I would like to be able to change the default CSS used by chrome, such as removing the underlines for links, etc. However, I cannot find the default CSS file used by Chrome. Does anyone know where it is? Helpful flags not getting updated after getting 'close question' privilege Possible Duplicate:          I recently got the ability to cast close votes. So, even if I flag a question as 'not constructive' or 'not a real question', it is still considered as a close vote. What is strange is that my 'helpful flag' count has stopped increasing after this change. Or at least it doesn't increase in cases where my flagging was automatically changed to a close vote and the post actually was closed. Is this how it is supposed to be?   I am guessing that once the post gets closed, the moderators are probably not bothered about marking flags as helpful. The only problem is that the Deputy badge will be much more difficult to get now. Earlier all the flags were getting converted to helpful, but now they just go in as close votes and don't show up on my profile page. In either case, one coin flip resulted in a head and the other resulted in a tail. Why is {H,T} a different outcome than {T,H}? Is this simply how we've defined an "outcome" in probability?   My main problem with {H,T} being a different outcome than {T,H} is that we apply binomial coefficients (i.e. we count subsets of sets) in some common probability problems. But if we take {H,T} and {T,H} to be different outcomes, then our "sets" are ordered, but sets are by definition unordered...   I feel as though the fact that I'm confused about something so basic means that I am missing something fundamental. Any help or insight whatsoever is greatly appreciated!