Why can't we substitute $(x-2)^2=t$ in the Definite Integral $$\int_0^3(x-2)^2dx$$? And what about the same Substitution in the indefinite integral $$\int(x-2)^2dx$$
Please consider the integral:  $$\int_{-a}^{a}x^2dx=\frac{2a^3}{3}$$  I would like to know why I can't make the substitution:  $$u=x^2$$  When I make the substitution, the limits of the integral will be the same, and the integral itself will be zero, which is the wrong answer. So why does this simple change of variables not work as I have expected?  Please note that I do not want help solving the integral, I know how to solve it several ways. My question is why does this specific attempt at a solution not work?
How To Auto-Format / Indent XML/HTML in Notepad++ Flex: Database driven DataGrid: arrows disappearing How to get over "device or resource busy"? I've got to send a bulk email out to a bunch of people in reply to emails they sent me. I've got all the original emails stored in a single outlook folder. I want to extract all of the reply addresses from the emails in that folder so I can send an email to all of them.  The article  shows you how to export the FROM address of these emails, however a large amount of them came from a web service that sends emails 'on behalf of' the person I'm trying to mail to. So I need the reply-to address, and the export wizard doesn't do that.  Does anyone have a tip on how to do this? This is Outlook 2007 on Exchange 2010. Where is this picture from? How do I use Reverse Image Search to find the source of an anime/manga image? So I see an image that I believe is from either an anime or a manga. What are some ways that I can go about finding what series it's from? Try these options:   (browser built-in support and plugins)        Note: Please note that as of May 9, 2016, ALL  questions are now . Therefore, you can't ask an identification request question on this site. However, such questions are still welcome  for users with at least  (globally on StackExchange).    Protip: If you are using Google Image Search in a non-English language,  or just use  for better, more accurate results. I'm a little confused as to when to use  for my physics class. For example, I'm asked to find the average speed of a race car that travels around a circular track with a radius of $500~\mathrm{m}$ in $50~\mathrm{s}$.  Would I need to apply the rules of significant figures to this step of the problem? $$ C = 2\pi (1000~\mathrm{m}) = 6283.19 $$  Or do I just need to apply significant figures at this step? $$ \text{Average speed} = \frac{6283.19~\mathrm{m}}{50~\mathrm{s}} = 125.664~\mathrm{m}/\mathrm{s} $$  Should I round $125.664~\mathrm{m}/\mathrm{s}$  to  $130~\mathrm{m}/\mathrm{s}$ since the number with the least amount of significant figures is two? parsing a date string from FTPClient.getModificationTime() Login not redirecting back to page where login link was clicked if account created What are the alternatives to Socket Programming? I was browsing around for programming troubleshooters, and I was wondering what the alternate would be for Socket Programming; if there is one. Can Windows Virtual PC run a 64-bit OS? What is the terminal's graphic API? How can certain programs set background and font colors, write characters to the same location on screen (e.g. to the third row, fifth column) more than once?  Some examples are the progress bar displayed by curl google.com &gt; a, some installation screens and nearly any text editor. Sudo is also user login password - why? how to change? When installing Ubuntu I can create a user with an encrypted folder, what I always do. Now after the installation the login password for this user is also the sudo password. Is that how it should be? I actually would like to have a different user login and sudo password, does is make sense? Should I do that, how? Export beamer slides to powerpoint/openoffice-impress/keynote editable format I usually prepare my talks using powerpoint/openoffice-impress/keynote, where I explaing programs, pseudocodes, etc. Recently I was thinking about creating text-only, code-only and pseudocode-only slides with beamer and then exporting them somehow to powerpoint where I modify them to add figures or some minor details for polishing the presentation, which are usually hard to do with beamer. I googled a bit around and found that beamer can export to PDF or PS but not to powerpoint. I wonder anyway if you know some possibility of conversion from beamer to on-screen editable powerpoint/openoffice-impress/keynote. Converting MultiLinestring to Linestring with PostGIS How to determine the amount of experience needed for leveling up Related questions under this topic might include:   How should I base how much experience points it takes to get to the next level? What factors are involved in this? How do I keep the player not entertained long enough till he levels up? How can I vary the way the player gets experience points? Formal basis for variable substitution in limits How to get icons in Safari favorites (aka favicons) to display correctly? I am using Safari on macOS Sierra. I have added several commonly-used sites to my favorites section, like so:    (there are 10 icons, and all but 2 of them are displayed as the proper, correct icon from the site. For example, the Wikipedia icon has the large W icon. However, for Google Drive and Gmail, the icons are just big, ugly, generic G's)  I would like for the two Google sites to correctly display their icons as well. At first, I thought it was because all 8 of the "correctly displayed" sites had successfully configured their "Safari settings" or something like that, while Google failed to do this (although this doesn't sound too plausible).  However, if I instead go to the top menu in Safari, click Bookmarks > Show Bookmarks, I get the following:    Strangely, this time, the two Google sites do have correctly displayed icons, while it is instead three other sites (out of the 10) that aren't displayed: namely Wikipedia, Amazon, and New York Times.  So it seems that there are two sets of icons, "big" and "small", and for big icons, Google Drive and Gmail are not displayed correctly, while the other 8 are, and on the other hand, for small icons, Google Drive and Gmail are displayed correctly, while Wikipedia, Amazon, and NYTimes are not.  I actually don't care too much about the small icons, but I would really like for all the big icons to display correctly. How can I do that? launchd plist format for running a command at a specific time on a weekday Rigorous proof that $dx dy=r\ dr\ d\theta$ I get the graphic explanation, i.e. that the area $dA$ of the sector's increment can be looked upon as a polar "rectangle" as $dr$ and $d\theta$ are infinitesimal, but how do you prove this rigorously?    1. Geometrically, the exact area would be $$\frac{(r+dr)^2d\theta}{2} - \frac{r^2d\theta}{2}$$$$= (r + \frac{dr}{2}) dr d\theta  $$$$= r dr d\theta + \frac{dr^2 d\theta}{2}.$$  How do we get rid of $\frac{dr^2 d\theta}{2}$? Is it too insignificant in value compared to $r dr d\theta$ so that it vanishes? If so, is it the reason it gets ignored? And what's more, geometrically speaking, why would the two areas - $dx dy$ and $r dr d\theta$, be equal?  2. Approaching it algebraically, setting $$x = r\sin\theta$$$$y = r \cos\theta$$  gives $$\frac{dx}{d\theta} = r\cos\theta,  dx = r\cos{\theta} d\theta$$$$ \frac{dy}{dr} = \cos\theta,  dy = \cos\theta dr. $$  Multiplying both equations, side by side, gives $$dxdy = r\cos^2\theta dr d\theta.$$  Again I get an extra term, which is $\cos^2 \theta$. In both cases I am unable to derive that $dxdy = rdrd\theta$.  What do I do wrong in my reasoning? It all makes me think that I'm getting something essential terribly wrong. I've just upgraded my computer hardware(cpu + motherboard + graphic card + memory + hard disk), so that install a new OS is needed. I tried to download debian-6.0.6-amd64-netinst.iso with wget command but the speed is so slow that I could not bear. 4Kb/s ~ 17 Kb/s, slow like a running turtle, or even more slower if I use Chrome.  I've read the help information of wget, it seems like there are no options could make it more faster.  Is there anyway to make wget faster? Or is it possible to make it multi-threading download?  PS: my bandwidth is 4M. I use this command:  wget -c url  http://hammurabi.acc.umu.se/debian-cd/6.0.6/amd64/iso-cd/debian-6.0.6-amd64-netinst.iso