Finding the integral of $\sin(x)\,dx$ using Riemann sums?
$\int_{\frac{1}{3}\pi}^{\frac{2}{3}\pi} {\sin(x)\;dx}$ using Riemann sums?
I have a motherboard with a PCIexpress x16 slot.  I want to purchase a card called AMD Radeon HD 7770, which uses PCIe version 3.0.  Is that backwards compatible? How to evenly space vertices? I have a edge loop that has vertices on it. Is there a way I can separate the vertices on the the edge evenly without moving the end points? I would also like for the vertices to maintain the current shape. Captchas not displaying in SO for me When I get a captcha on Stack Overflow, I don't see the image or the input box with Firefox. I originally had the "HTTPS Anywhere" extension installed but still have the same issue when it's disabled (This caused issues previously when Google's SSL certs were expired for an old domain IIRC).  The requests to Google seem to be never-ending; browsing to the URLs that are still loading results in some pages of JavaScript, but no errors as you'd hope.  The screenshot was taken after the page had been open for more than 3 minutes, and I'm not sure why the GETs are still running. The request popup visible in Firebug is for "GET challenge?k=...."  When browsing to the same captcha page in IE, there are no issues.  I'm aware this could be an issue with my Firefox configuration, as opposed to the site, but would appreciate it if someone could verify if it's just me, and suggest a fix.  EDIT   While visiting the Recaptcha site, I can see a Captcha with no problems If I right-click the problem URLs in FF and Open in new tab, the pages load fine. From the fact that FB omits IP Addresses for the problem ones, I assume that it's not even opening a connection (nor perhaps performing a DNS lookup) Other sites that use Recaptcha seem to work fine (I picked 3 at random from Google) I would like to remove page numbering on a specific page. However, the header and footer should remain intact. If I try:  \thispagestyle{empty}   The header and footer disappear. What is the recommended way of removing just the page number? Why doesn't hydrogen gas exist in Earth's atmosphere? Grep: unexpected results when searching for words in heading from man page I'm running into weird behavior when trying to grep a man page on macOS. For example, the Bash man page clearly has an occurrence of the string NAME:  $ man bash | head -5 | tail -1 NAME   And if I grep for name I do get results, but if I grep for NAME I don't:  $ man bash | grep 'NAME' $ man bash | grep NAME   I've tried other uppercase words that I know are in there, and searching for SHELL yields nothing whereas searching for BASH yields results.  What's going on here?  Update: Thanks for all the answers! I thought it worth adding the context in which I ran into this. I wanted to write a bash function to wrap man and in cases where I've tried to look up the man page for a shell builtin, jump to the relevant section of the Bash man page. There might be a better way, but here's what I've got currently:  man () {   case "$(type -t "$1")" in     builtin)       local pattern="^ *$1"        if bashdoc_match "$pattern \+[-[]"; then         command man bash | less --pattern="$pattern +[-[]"       elif bashdoc_match "$pattern\b"; then         command man bash | less --pattern="$pattern[[:&gt;:]]"       else         command man bash       fi       ;;     keyword)       command man bash | less --hilite-search --pattern='^SHELL GRAMMAR$'       ;;     *)       command man "$@"       ;;   esac }  bashdoc_match() {   command man bash | col -b | grep -l "$1" &gt; /dev/null } Var[A+B+C] = Var [A] + Var [B] + Var [C] I know that this relationship is correct when A,B and C are pairwise independent. However, I'm finding it difficult to calculate it.  Let's say that: Var [A] = Var [B] = Var [C] = 0.25. How can I prove that Var[A+B+C] = 0.75?  I started to work with the formula: Var[A+B+C] =E[(A+B+C)^2]−(E[A+B+C])^2 but I got stuck.  Thanks! A group $G$ with $G'$ abelian and every abelian normal subgroup finite Let $G$ be a group such that  $G'$ abelian  and any abelian normal subgroup of $G$ is  finite. Show that $G$ is finite. Why do shields in Star Trek "wear out"? Possibilities for Python classes organized across files? I'm used to the Java model where you can have one public class per file.  Python doesn't have this restriction, and I'm wondering what's the best practice for organizing classes. Why isn’t the definite article used in "The commission commanded that work [...] should cease"? In this sentence:     The commission commanded that work on the building should cease.   Why isn't the used before the word work? It seems that the work is specified by on the building.  Another example is,     Construction of the building will start next week.   I understand construction here is something like putting up the building, and so the construction isn’t general in meaning. But the definite article isn’t used here, either.  I’ve asked about this several times, but none gave me a good answer. Convergence of Ratio Test implies Convergence of the Root Test In Elias Stein and Rami Shakarchi's Complex Analysis textbook, we have the following exercise:     Show that if $\{a_n\}_{n=0}^\infty$ is a sequence of complex numbers such that $$\lim_{n\to\infty}\frac{|a_{n+1}|}{|a_n|}=L,$$ then $$\lim_{n\to\infty}|a_n|^{1/n}=L.$$    I've been trying to prove this with no luck. The only thing I've thought of doing is $$\lim_{n\to\infty}\left(\frac{|a_{n+1}|^n}{|a_n|^n}\right)^{1/n},$$but this hasn't lead me anywhere except dead ends. Will someone provide a hint for me about how to proceed? Thanks!  Minor update: I don't know if it's helpful yet, but I know we can write the limit as $$\lim_{n\to\infty}\left(\frac{|a_{n+1}a_n\cdots a_0|}{|a_n\cdots a_0|}\cdot\frac{1}{|a_n|}\right).$$This reminds me a lot of the geometric mean, which even has the exponents I'm trying to get... latexmk: jobname option and clean up incompatibility? I'm using  in a Makefile to compile my document. Since I want my output filename to be different from the input filename, I use latexmk's option jobname.  The problem is: it seems to be incompatible with latexmk -c to clean temporary files.  Here's the specific entry in my Makefile:  these.pdf: header.tex $(OBJS)     latexmk                 \         -pdf                \         -pdflatex="$(TEX)"  \         -jobname="these"    \         -use-make $&lt;   Input files are header.tex containing the latex instruction header and other latex content files. The output files are these.{aux,bbl,bcf,...}. Then latexmk -c does not remove temporary files: .{aux,bbl,bcf,...} files. Why is a different dll produced after a clean build, with no code changes? When I do a clean build my C# project, the produced dll is different then the previously built one (which I saved separately). No code changes were made, just clean and rebuild.   Diff shows some bytes in the DLL have changes -- few near the beginning and few near the end, but I can't figure out what these represent. Does anybody have insights on why this is happening and how to prevent it?   This is using Visual Studio 2005 / WinForms.  Update: Not using automatic version incrementing, or signing the assembly. If it's a timestamp of some sort, how to I prevent VS from writing it?   Update: After looking in Ildasm/diff, it seems like the following items are different:   Two bytes in PE header at the start of the file. &lt;PrivateImplementationDetails&gt;{guid} section  Cryptic part of the string table near the end (wonder why, I did not change the strings) Parts of assembly info at the end of file.   No idea how to eliminate any of these, if at all possible... Feeling a tingling sensation by touching stuff, Is this safe? I'm putting on my desktop computer on steel rack as usual. A while ago, when I touch the steel rack for seconds, practically I feel a tingling sensation, sort of electrical shock in my finders. After that I measured how voltage the steel rack has, the multi-tester meter indicated almost 50V by AC mode. Is this safe to touch or keep the PC on the position, or I should get away the PC somewhere electricity-safe? I'm not sure this site is suit for asking this kind of question, but I'm very worrying whether this is safe or not since this experience is my very first time. Thank you. I have 2 storage devices; classical slow HDD (750GB, /dev/sda) and faster SSD (128GB, /dev/sdb). Currently I have installed Ubuntu &amp; Mint on same btrfs partition on SSD (/dev/sdb5). My btrfs pool consists of /dev/sdb5.  What I want to achieve is data replication on HDD while keeping SSD performance: Something like this:  For read case it would be ideal for procedure to go like this:   read from SSD check checksum if it verifies.. awesome, give me the data if not, fetch data from HDD and perform error correction through duplicate data   I'd manually run (some tool I'm not sure which one... btrfs check --repair perhaps?) every month or so to check HDD checksums and correct them via SSD data against bitrot.  For write case:   write to SSD sync to HDD when write requirements are low, thus not slowing down the system.   Is this possible and how would I do it?  Bottom line is, I like btrfs and would like it to duplicate data across HDD &amp; SSD drive automatically while keeping SSD excellent performance.   Using SSD as write cache is not an option since it's missing data duplication and security. Evaluate $\lim_{n\rightarrow\infty} \int_0^1 f(x^n) dx$ Let $f$ be a real, continuous function defined for all $0\leq x\leq 1$ such that $f(0)=1$, $f(1/2)=2$, and $f(1)=3$. Show that  $$\lim_{n\rightarrow\infty} \int_0^1 f(x^n) dx$$  exists and compute the limit.  Attempt:   Since $f$ is real-valued continuous on $0\leq x\leq 1$ and the boundaries $f(0)=1$ and $f(1)=3$, $f$ is bounded on the interval  and the integral $\displaystyle\int_0^1 f(x^n)dx$ exists for any positive $n$. Thus, we can interchange the limit and the integral and compose the limit, \begin{align*} \lim_{n\rightarrow\infty} \int_0^1 f(x^n)dx&amp;=\int_0^1\lim_{n\rightarrow\infty}f(x^n)dx\\ &amp;=\int_0^1 f\left(\lim_{n\rightarrow\infty} x^n \right)dx\\ &amp;=\int_0^1 f(0)dx=1. \end{align*}  Questions: So, my first question concerns whether interchanging the limit and the integral is correct. It seems it would be justified by the dominated convergence theorem, where $f(x^n)$ is dominated by a function $g(x)=\displaystyle\sup_{0\leq x\leq 1} f(x^n)$ and $f_n(x^n)$, on $0\leq x\leq 1$, converges pointwise to a function $f$ that takes value 1 for $x\neq 1$ and 3 for $x=1$.  My next question is whether the interchange between the limit and composition of the function is allowed. Since the function is continuous and the limit exists at that point, it seems the interchange would be justified. Why does Python's itertools.cycle need to create a copy of the iterable? Black and white beads on a circle There are $n$ beads placed on a circle, $n\ge 3$. They are numbered in random order as viewed clockwise. Beads for which the number of the previous bead is less than the number of a next bead are painted in white color,and others - in black.  Two colourations that can be made equal by rotation are considered identical. How many different colourations can occur?  I've write a programm and for $n=3...11$ I've got answers $2 , 1 , 6 , 7 ,  18 , 25 , 58 , 93 , 186$ "object-oriented" vs "object oriented" When should we use "object-oriented" and when "object oriented" when talking about programming?  An example:     Why should I start writing object oriented code?   I personally think that there should be a hyphen, because "oriented" connects to the word "object", but I am not sure.