For my calculus class, one of the review questions I'm given is  Find the volume V by rotating the region bounded by $y = 5x-x^2$ and $y = x^2 - 5x + 8$ about the $y$-axis.  I've never learned this concept before in high school or previous mathematics courses, can someone explain how to do this? Would appreciate any help!
Find the volume of rotation about the y-axis for the region bounded by $y=5x-x^2$, and $x^2-5x+8$  Here is an image:    Normally I can do this question, but this one is tricky because since we are rotating about the y-axis, and we are quadratic, when I solve for $x$ I get two answers, one positive and one negative.   What I mean is that if you look at the graph $5x-x^2$, then there are two halves, and if you look at the green line, that's one half of the curve (positive one), and the red half is the negative one.  Since we can see both the negative and positive sections in this graph, and both are being rotated, I wasn't sure how to create the equation that I can later integrate.  Usually in the problems I do, the negative doesn't exist, and so it's trivial. In this case I am not sure.
Starcraft Installation asks for a mythical Disk 2 I just purchased the Starcraft Battle Chest containing Starcraft and BroodWar.  When I try to install Starcraft I get halfway through and I'm prompted to "Insert Disk 2".  The box did not contain a second disk.  My computer is Windows 7 (x64). What do I need to do to continue?  Box Contents:   Starcraft Disk Broodwar Disk WoW Trial Disk Cross product: matrix transformation identity How can one prove the following identity of the cross product?  $$(M a)\times (M b)=\det(M) (M^{\rm T})^{-1}(a\times b)$$  $a$ and $b$ are 3-vectors, and $M$ is an invertible real $3 \times 3$ matrix. Short way for upper triangularization How to visualize a value resulted from math node? How to visualize a value resulted from math node,or just to know it. I tried the way of scaling this value to the scene size and then pick the color and get the value but I want more effective way to do this. Let $f: B(0,1) \rightarrow \mathbb{C}$ be holomorphic and suppose $\exists\ r \in (0,1)$ such that $f$ is injective in $A = \{z \in \mathbb{C} : r &lt; |z| &lt; 1\}$. Prove that $f$ is injective.  I tried using Rouché theorem, or the identity theorem, but I don't know what to do. Any hints? :) Diamond moderator privileges vs. 10k rep privileges? As we are in election period, I don't get the difference between getting to 10k reputation point and the diamond. What additional controls do you earn with the diamond ? Intuitive Understanding About the Implicit Function Theorem In multivariable calculus, given a function like $F(x,y,z) =0$, the implicit function $z=f(x,y)$ exists if and only if $\frac{\partial F}{\partial z} \not = 0$. And the implicit function is given by $\frac{dz}{dy}=-\frac{\frac{\partial F}{\partial y}}{\frac{\partial F}{\partial z}}$.   The theorem can be proved by mathematical reasoning. But I want to know whether there is some intuitive understanding of the theorem, say geometry intuition in the three dimensional space, etc. novel with psi slavery Why does Worf want O'Brien to go to Minsk? Subspace of Noetherian space still Noetherian I have a script which calls two commands:  long_running_command | print_progress   The long_running_command prints progress but I'm unhappy with it. I'm using print_progress to make it nicer (namely, I print the progress in a single line).  The problem: Connection a pipe to stdout also activates a 4K buffer, so the nice print program gets nothing ... nothing ... nothing ... a whole lot ... :)  How can I disable the 4K buffer for the long_running_command (no, I do not have the source)? How do I find a command if I don't know its name? How to redirect output to a file and stdout In bash, calling foo would display any output from that command on the stdout.  Calling foo &gt; output would redirect any output from that command to the file specified (in this case 'output').  Is there a way to redirect output to a file and have it display on stdout? How to set a short password on Ubuntu? Permission denied in 777 folder john@ubuntu01:~$ sudo ls -l /var/opt/gitlab/git-data/repositories/GL/www.git total 25068 drwxr-xr-x  2 git  git      4096 aug 14 19:58 branches -rw-r--r--  1 git  git        66 aug 14 19:58 config drwxrwxrwx  2 root root     4096 aug 15 14:56 custom_hooks -rw-r--r--  1 git  git        73 aug 14 19:58 description -rw-r--r--  1 git  git        23 aug 14 19:58 HEAD lrwxrwxrwx  1 git  git        47 aug 14 19:58 hooks -&gt; /opt/gitlab/embedded/service/gitlab-shell/hooks drwxr-xr-x  2 git  git      4096 aug 14 19:58 hooks.old.1471193907 -rw-r--r--  1 git  git      1329 aug 15 14:48 index drwxr-xr-x  2 git  git      4096 aug 15 13:58 info drwxr-xr-x 71 git  git      4096 aug 15 14:55 objects -rw-r--r--  1 git  git        98 aug 15 13:58 packed-refs -rw-r--r--  1 git  git  25618530 aug 15 14:03 post-receive.log drwxr-xr-x  5 git  git      4096 aug 14 20:31 refs john@ubuntu01:~$ ls -l /var/opt/gitlab/git-data/repositories/GL/www.git ls: cannot access '/var/opt/gitlab/git-data/repositories/GL/www.git': Permission denied   Here are ACLs  john@ubuntu01:~$ sudo lsattr /var/opt/gitlab/git-data/repositories/GL/www.git/custom_hooks -------------e-- /var/opt/gitlab/git-data/repositories/GL/www.git/custom_hooks/post-receive   I have never seen this before. No permission to LS even with 777? I need to share this folder for more convenient development. How can I solve it? Thank you.  EDITED: (here are your requests)  john@ubuntu01:~$ sudo ls -ld /var/opt/gitlab/git-data/repositories/GL/www.git drwxrwxrwx 8 git git 4096 aug 15 14:48 /var/opt/gitlab/git-data/repositories/GL/www.git john@ubuntu01:~$ sudo ls -ld /var/opt/gitlab/git-data/repositories/GL/www.git/custom_hooks drwxrwxrwx 2 root root 4096 aug 15 14:56 /var/opt/gitlab/git-data/repositories/GL/www.git/custom_hooks   EDITED MORE  sudo ls -ld /var/opt/gitlab/git-‌​data/repositories/GL/‌​www.git ls: cannot access '/var/opt/gitlab/git-‌​data/repositories/GL/‌​www.git': No such file or directory   hmmm... what is this? Is it okay to publish my personal notes from a lecture? How do I counter-knife an enemy? When you press down the right stick to counter the attack, do you just press it, repeatedly press it, or press and hold? I've been knifed a few times from the front and each time I try to counter I fail. Also, if you believe you are about to be knifed and hold down the right stick until you are knifed, will that counter an attack, or do you have to press it down just at the right time? "It was my op" what does it mean? I heard this in  movie. It was my op. What does it mean? Googling gave me : 'Original Poster'. But that doesn't seem very meaningful. When compiling a beamer presentation and using the following \author command  \author{Name \\ \texttt{my.email@domain.com}}   I get the following hyperref warning in my logfile  Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref)                removing `\\' on input line 15.   Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref)                removing `\new@ifnextchar' on input line 15.   I understand that this has to do with hyperref setting the PDF metadata, where the linebreak does’t make much sense and should be removed.  Trying to set  \hypersetup{pdfauthor={Name}}   does’t change the situation, seems like hyperref is still looking at the author command.  How do I get the desired display of author name with email and still keep hyperref happy? Study the following limit: $$ \lim\limits_{ (x,y,z) \to (0,0,0) } \frac{xy-z^2}{x^2+y^2+z^2} $$   I'm not sure if this way of proving that the limit does not exist is correct so I'm this is basically for a proof verification.  Approaching $(0,0,0)$ through the plane $z=0$, then the limit becomes  $$ \lim\limits_{ (x,y) \to (0,0) } \frac{xy}{x^2+y^2} $$  Now this limit does not exist, because approaching $(0,0)$ through the line $y=0$, the limit becomes  $$ \lim\limits_{ x \to 0 } \frac{0}{x^2} =0$$  But if we approach $(0,0)$ through the line $y=x$ we have that  $$ \lim\limits_{ (x,x) \to (0,0) } \frac{x^2}{2x^2}=\lim\limits_{ (x,x) \to (0,0) } \frac{1}{2} =\frac{1}{2}$$  So the limit $ \lim\limits_{ (x,y) \to (0,0) } \frac{xy}{x^2+y^2} $ doesn't exist. This means that the limit we began with doesn't exist, correct?