Why is the surface area of a revolution $\int 2\pi y ds$ instead of $\int 2\pi y dx$? I learned from this  that the surface area of a revolution is $\int 2\pi y ds$. Why is it not $\int 2\pi y dx$? I know this is a , but I do not think I get a satisfactory answer. There is another  that told us the area under curve is $\int y dx$. Why are we using $dx$ in the area calculation, but not the surface area of a revolution? Why can we use the argument that when the $\Delta x$ gets smaller and smaller, the limitation of $\sum y \Delta x$ is the area under curve, but we cannot use the same argument to the surface of a revolution? Why not say when the $\Delta x$ gets smaller and smaller, the limitation of $\sum 2\pi y \Delta x$ is the surface area of the revolution?
Areas versus volumes of revolution: why does the area require approximation by a cone? 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.
Position of a relative clause before/after a verb "A friend of my father" vs. "a friend of my father's" — which one is right? Why? I have seen them both in my reference book, which is written by a native speaker. I wonder why there are two different phrases to describe the same meaning and am looking forward to some reasonable details to explain why this is so. I've checked  for acute:  an acute illness is one that has quickly become severe and dangerous Compare:CHRONIC  And for chronic:  lasting for a long time Compare:ACUTE  Can a disease be both &quot;acute&quot; and &quot;chronic&quot; simultaneously? For example it could be the case where a certain illness quickly becomes serious and have long term consequences Are there any &quot;obtuse illnesses&quot; ? I want to know where I can find updated information about all the stable and in-development versions of Ubuntu, where I can find this information?     This question is meant to be a canonical question. It expects to answer the following questions:         Is X version stable yet?   Is X version in beta?   Is X version still supported?   What versions are still supported?   What are the versions in Beta/Alpha?   Until where X version will be supported?   Have X version ceased support?   When will be X version released?   Where I can download any release?   When will the next LTS version of Ubuntu be released?         The list is not exhaustive. Is there any way in QGIS to apply service area buffers e.g if I have a point on a map I wish to show the service coverage of 1 mile, 2 mile, 3 mile, breaks using  the road network? How can I sort a string of text followed by a number using LINQ I heard that there was a twitter bot which automatically posted questions and asked for answers for specific questions on every site. How do they work? Is there an algorithm for doing this or is it purely random? I know that it doesn't tweet closed questions, but that's it. References to a comment on a deleted answer can break Stack Exchange JS For the following URL, JavaScript breaks:  https://stackoverflow.com/questions/16356382/splitting-a-braces-grouped-string-in-python/16356421?noredirect=1#comment23433042_16356421   This is a link to a , the answer I can see as a 10K+ user (although users &lt;10k ).  With Chrome (26, Mac OS X) the console sez:  Uncaught TypeError: Cannot call method 'replace' of undefined   for line 83 of full.js.  Firefox (20, Mac still), the web console spits out:  [12:48:28.459] TypeError: c.attr(...) is undefined @ http://cdn.sstatic.net/js/full.js?v=c9131e40e3fb:83   Safari (6.0.4 on, you guessed it, Mac OS X) has this to complain about:  TypeError: 'undefined' is not an object (evaluating 'c.attr("id").replace')   for line 83 of full.js.  Because of this, the Answer text box on Stack Overflow is lacking all niceties such as an actual WMD editor.  If you remove the hash reference #comment23433042_16356421 from the end of the URL, things work again. Removing the ?noredirect=1 query parameter makes no difference. Internal keyboard and trackpad stop working after plugging in USB device I got a new MacBook Pro with OS X Mavericks. When connecting my phone via USB, the internal keyboard and the trackpad immediately stop working -- not always but in most cases. The bluetooth mouse and an external USB keyboard are not affected. Everything else works also fine. The power button of the internal keyboard still works. Reboot helps. Any ideas? How do I URL encode a string Why doesn't Volumetric follow the shape of Mesh in Eevee? So I was in the middle of making a ball-shaped smoke, but the volumetric effect didn't work following the basic shape. It remains cube in Eevee Blender 2.8; while in Cycle it works, but in Cycle it takes thousands of years for Render. How to send data to a serial port and see any answer? On Linux, I want to send a command string (i.e. some data) to a serial port (containing control characters), and listen to the response (which also usually might contain control characters).   How can I do this as simplest as possible on Linux? An example is appreciated! download.01.org certificate verification failed - CRLfile missing? So I had intel-Linux-graphics-installer on my computer  but lately when I   sudo apt-get update   I get the following error  Err https://download.01.org trusty/main amd64 Packages                          server certificate verification failed.  CAfile: /etc/ssl/certs/ca-certificates.crt CRLfile: none   Tried reinstalling intel-linux-graphics-installer but nothing happend. And various solutions found here on askubuntu.com on certificates problems but didnt end up anywhere.  Any ideas on how to get that CRLfile? Also is this problem on my side or on the servers side? How do I sort a vector of pairs based on the second element of the pair? If I have a vector of pairs: std::vector&lt;std::pair&lt;int, int&gt; &gt; vec;  Is there and easy way to sort the list in increasing order based on the second element of the pair? I know I can write a little function object that will do the work, but is there a way to use existing parts of the STL and std::less to do the work directly? EDIT:  I understand that I can write a separate function or class to pass to the third argument to sort.  The question is whether or not I can build it out of standard stuff.  I'd really something that looks like: std::sort(vec.begin(), vec.end(), std::something_magic&lt;int, int, std::less&gt;()); I installed CCSM to enable window previews in Unity. But when I went into extras, there was no 'window preview' option.  So I tried experimenting with other options in Compiz and now there are no borders in the dialogue boxes (close, minimize, restore buttons are not there, I had enabled some 'flat....' option in Compiz.  Later, I uninstalled Compiz and then reinstalled it. Now it says      "Failed to load session Ubuntu"   I can't even access through the guest account, I am new to Ubuntu.(please explain in detail).  All I have access to is the login screen (it still shows my custom wallpaper). Please help, I don't want to lose my stuff. Why are there so many bottomless pits in Star Wars How do I identify SMD components? (or how do I identify any component) I have a laptop which I'm going to return due to battery defects. As I have already configured everything on this laptop, I'd like to migrate the installation on the new laptop (when I'll get it). The one to replace has a 128 GB eMMC, and two USB 3 ports. So, I was thinking to   Boot from a Linux live USB from port 1 Make an image with dd of the eMMC, saving it to a external HDD on port 2 Do the reverse on the new laptop   My question is, will it work fine with Windows? Does migrating this way have any issues? Would I be able to restore the installation if the new laptop is a different model with at least 128 GB? Find all roots of $\,x^2\!\equiv x\pmod{\!900}$, i.e all idempotents in $\Bbb Z_{900}$ I need the idempotent elements of $Z_{900}$ $2^2\cdot 3^2\cdot 5^2=900$ Of course there's $$0 \pmod 4 \\ 0 \pmod 9 \\ 0 \pmod {25} \\ $$ and $$ 1 \pmod 4 \\ 1 \pmod 9 \\ 1 \pmod {25} \\ $$ I found the answers by making a C++ program to test all numbers, but I don't know how to quick solve on paper. This is the output of the program $(0, 0, 0) \rightarrow 0 \\ (1, 1, 1) \rightarrow 1 \\ (0, 0, 1) \rightarrow 576 \\ (0, 1, 0) \rightarrow 100 \\ (1, 0, 0) \rightarrow 225 \\ (1, 1, 0) \rightarrow 325 \\ (0, 1, 1) \rightarrow 676 \\ (1, 0, 1) \rightarrow 801 \\$ This is a question that's been nagging me for some time. I always thought that C++ should have been designed so that the delete operator (without brackets) works even with the new[] operator.  In my opinion, writing this:  int* p = new int;   should be equivalent to allocating an array of 1 element:  int* p = new int[1];   If this was true, the delete operator could always be deleting arrays, and we wouldn't need the delete[] operator.  Is there any reason why the delete[] operator was introduced in C++?  The only reason I can think of is that allocating arrays has a small memory footprint (you have to store the array size somewhere), so that distinguishing delete vs delete[] was a small memory optimization.