When deriving the formula for the area of a solid rotated about an axis, my textbook (Stewart calculus) uses portions of circular cones and integrates their lateral areas. I understand this process.  My question is why doesn't integrating the lateral areas of circular cylinders work as well. After all, we can use rectangles (and not trapezoids) to evaluate areas under curves, using cylinders as opposed to cone segments seems like the three dimensional equivalent.  I've been through both processes and I see why they don't give the same formula, but conceptually I don't understand why cylinders are wrong.  Thanks a lot
Suppose we rotate the graph of $y = f(x)$ about the $x$-axis from $a$ to $b$.  Then (using the disk method) the volume is $$\int_a^b \pi f(x)^2 dx$$ since we approximate a little piece as a cylinder.  However, if we want to find the surface area, then we approximate it as part of a cone and the formula is $$\int_a^b 2\pi f(x)\sqrt{1+f'(x)^2} dx.$$  But if approximated it by a circle with thickness $dx$ we would get $$\int_a^b 2\pi f(x) dx.$$  So my question is how come for volume we can make the cruder approximation of a disk but for surface area we can't.
Let $$f(x)=\begin{cases} \sin \tfrac 1 x &amp;\text{if $x\ne 0$}\\ 0 &amp; \text{if $x = 0$.}\end{cases}$$ I have to show that $f$ has the intermediate value property. That is, for any $a &lt; b$, if $y$ is any real number such that $f(a) &lt; y&lt; f(b)$ or $f(a)&gt;y&gt; f(b)$, then there exists a $c \in (a,b)$ such that $f(c)=y$.  I feel like I kind of know how to go about completing this. I just am curious as to if I have to create a bound such as letting $a = -1$ and $b = 1$, or keep $a$ and $b$ both arbitrary. On the Android app, could we have the ability to flag comments we've previously upvoted? This is a fairly regular need for me, and I ran into it again earlier: I flag comments as obsolete when they are, and fairly often, one of them will be a comment I've upvoted.  For an example of a typical case: a comment suggests a great improvement to a question, I upvote it, the author performs that improvement, the comment is now obsolete and (on my PC) I flag it as obsolete. The comments get cleaned up and the question moves on.  There's a request that's been around since 2011 for the site itself: . It's implemented by SOUP (as described ), but the Android app doesn't have a SOUP available to add this functionality to the app for me. Reasons why not to update an Ubuntu version to a newer/older kernel version Review sections to be always in the same place Boot Camp error "Windows cannot locate the disk and partition" I have a 2019 MacBook Pro 16 inch running on macOS Big Sur 11.1. I am trying to install windows using Bootcamp. I get the following error: Windows cannot locate the disk and partition specified in the unattended answer file's  setting. Make sure the setting references a valid partition and restart the installation. Can someone help me how to solve this? Thanks in advance Cartesian product of two $\sigma$-algebras is generally not a $\sigma$-algebra? When reading the book Measures, Integrals and Martingales written by R.L. Schilling, I saw a statement as below: $$\mathcal{A} \times \mathcal{B}:= \{A \times B: A\in \mathcal{A}, B\in \mathcal{B}\}$$ where $\mathcal{A}$ and $\mathcal{B}$ are $\sigma$-algebras, is not $\sigma$-algebra in general.  However, I cannot construct a counterexample. Could anyone offer help here?  Kind regards In my special package combination I get the error No room for new \write. \tablecontents and Bad number (16). \tablecontents. The following MWE throws this error in pdflatex and lualatex with TeX Live 2015. As you can see, I add the own content register loi. I can delete nearly every package separately and the error directly vanishes.  Does anyone have an idea what is wrong with this package combination?  \documentclass{scrartcl} % load packages \usepackage{ifluatex} \ifluatex     \usepackage{fontspec} \fi \usepackage{adjustbox, amsmath, blindtext, datatool, fancybox, fixme, graphicx, marginnote, pgfplots} \usepackage{imakeidx} \usepackage{biblatex} \usepackage{glossaries} \usepackage{hyperref} % load package for code code highlighting \usepackage{minted}  % define Open Issue and corresponding List of Open Issues \newcommand{\openissue}[2][Unlabeled]{%     \par%     {\refstepcounter{loicounter}%     \phantomsection% comment out if hyperref is noy used     \addcontentsline{loi}{figure}{%         \protect\numberline{%             \ifcsname c@chapter\endcsname%                 \thechapter.%             \fi%             \theloicounter%         }{#1: #2}%     }%     \textbf{Open Issue -- #1:} #2}%     \par% } % define counter \makeatletter \ifcsname c@chapter\endcsname%     \newcounter{loicounter}[chapter]% \else     \newcounter{loicounter}% \fi% \newcommand*{\listopenissuesname}{List of Open Issues} \newcommand{\listofopenissues}{%     \ifcsname chapter\endcsname%         \chapter*{\listopenissuesname}%     \else%         \section*{\listopenissuesname}%     \fi%     \@starttoc{loi}% } \makeatother  \fxsetup{status = draft} \makeglossaries \makeindex  \begin{document}  % include open issues \listofopenissues % include list of corrections \listoffixmes % include indicies \tableofcontents                            % Inhaltsverzeichnis einbinden  \end{document} What would happen if you used a dead person's hair in Polyjuice Potion? Polyjuice Potion is changing based on someone's current physical state. It shows tattoos (or lack of) and such from DH when Ron changed to Harry.     "I knew Ginny was lying about that tattoo,” said Ron, looking down at his bare chest.    What would happen if someone used hair of a dead person, taken from their body after death, and drank the potion? Would they die as well? Not getting any reading from FTDI UMFT201XB USB-to-I2C module I got the  in order to interface with an I2C peripheral through a USB port on my Windows 7 computer.  The VCP drivers seem to install correctly, and a COM port is assigned to the module, and can be opened by a terminal emulator. However, I cannot seem to get any signal in or out of the module. That is, I see no activity when I send data through the port, or receive any data when I connect an I2C device to it (this was debugged using a scope). I have also verified that the RESET isn't on.  I found very little helpful information out there on how to get this module to work, so if anyone has experience with it, or with similar FTDI modules, my question is: what are the steps that I need to take in order to get the module to communicate? I'm trying to solve hard combinatorics that involve complicated factorials with large values.  In a simple case such as $8Pr = 336$, find the value of $r$, it is easy to say it equals to this: $$\frac{8!}{(8-r)!} = 336.$$  Then $(8-r)! = 336$ and by inspection, clearly $8-r = 5$ and $r = 3$.  Now this is all and good and I know an inverse function to a factorial doesn't exist as there is for functions like sin, cos and tan etc. but how would you possibly solve an equation that involves very large values compared to the above problem without the tedious guess and checking for right values.  Edit: For e.g. if you wanted to calculate a problem like this (it's simple I know but a good starting out problem) Let's say 10 colored marbles are placed in a row, what is the minimum number of colors needed to guarantee at least $10000$ different patterns? WITHOUT GUESS AND CHECKING  Any method or explanation is appreciated! I know questions on the "hot" tab of a site homepage are based on some sort of views/velocity criteria, but does anyone know the actual algorithm used?  Background: I was considering a feature-request about the "hot" tab, but I don't have enough information to speak about it intelligently. How do I pre-install Ubuntu for someone (OEM install)? I would like to pre-install Ubuntu on a computer for someone. I may want to customize it for them, but in the end they need to be able to choose a username, password, etc. when they first boot it up. (Also called an OEM install).  How do I do this? Can I play Minecraft Java if I only buy the Windows 10 version I do not own Minecraft outside my PS4, I know I can't play either Windows 10 or Java with that purchase. However, my primary computer is Linux and only supports the Java edition, and its past the deadline to get a redeem code for Windows 10 edition if I purchase the Java Edition.    Can I purchase the Windows 10 edition and use that purchase to login and play on the Java edition? I'm working through a problem regarding expected values in Markov chains, and at some point it says:     Recall from probability that if $X$ is a positive integer valued random variable, then $\mathbb{E}[X] = 1 + \sum^\infty_{k=1}\mathbb{P}(X &gt; k)$.   I know that by definition $\mathbb{E}[X] = \sum_a a\mathbb{P}(X = a)$, but I can't see how the above equality follows from this, nor am I sure if this is how to approach the problem. Can I change some but not all Apple keyboard function keys (dim vs F1, bright vs F2)? I want to override F1 and F2 behavior, since I have external display and Mac can't control display's brightness. The rest of the keys should retain their non Fx  I know about "Use all F1-F12 keys as standard function keys.", but I don't want to change all of the keys to be F when pressed, just some of them. (For me single press music and volume control is necessary.)  Is there a way to pick some of the function keys to default to numbered function instead of the "Apple" definitions system wide? How to explain the long lifetime of Rydberg atoms with Fermi golden rule? I'm new to blender and have been studying and starting with low poly modeling. Recently I changed my GPU from a GTX 960 to an RX 580. I uninstalled the drivers and installed new ones using DDU.  Problem: Rendering using OpenCL produces these unwanted lines. The attached image is a product of CPU + GPU render then the GPU only render, respectively. When I render using only the CPU the results are as expected, no unwanted black vertical lines, using only the GPU adds more lines on the image.  Any help is appreciated, thank you! I have a registered StackOverflow account which I normally log into using OpenID.  Recently I logged out in order to post a question anonymously (supplying a fake name and of my secondary email addresses).  The problem now is that I'm logged in to StackOverflow as this "dummy" user, unable to see a way to log out.  The top menu normally has a "logout" link, but when you aren't fully registered this link is replaced with "register".  There is no "logout" link for me. :(  So..   When logged out there is a "login" link. When logged in there is a "logout" link. When "logged in" with an unregistered account, there is only a "register" link (no "logout"). In addition to showing reputation scores, SO should rank users. In the users section, users are ordered by the reputations they have.  Each user should also have a unique rank, since many users can have the same reputation score.  Rank must follow the order laid out by reputation, but for those having similar reputation, an additional formula should be defined which can rank users based on tags, time since registration, frequency of activity and/or helpfulness of their answers.  What are your thoughts on this? Given a list ["foo", "bar", "baz"] and an item in the list "bar", how do I get its index (1) in Python?