Why does not $ \int_{b}^{a} 2\pi \,f(x) \,dx $ yield the correct answer when calculating the surface area of a solid of revolution?
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.
How to text-to-speech output using command-line? How to get speech output from entered text by using command-line?  Also facility to change speech rate, pitch, volume etc using simple command. How to assign values to a MATLAB matrix on the diagonal? $f(x)f(1/x)=f(x)+f(1/x)$ Find a function $f(x)$ such that:  $$f(x)f(1/x)=f(x)+f(1/x)$$  with $f(4)=65$.  I have tried to let $f(x)$ be a general polynomial: $$a_0+a_1x+a_2x^2+\ldots a_nx^n$$  which leaves $f(1/x)$ as:  $$a_0+a_1{1\over x}+a_2{1\over x^2}+\ldots + a_n{1\over x^n}$$  On comparing the coefficients of both sides, we see that:  $$2a_0=(a_0)^2+(a_1)^2+(a_2)^2+\ldots+(a_n)^2$$  And  $$a_1=(a_0a_1)+(a_1a_2)+ \ldots +(a_{n-1}a_n)$$  I don't know how to proceed further. I know I need to compare coefficients and come to a conclusion based on their values, but I don't see what to do next. Xubuntu live usb persistence - how to? I'm using Xubuntu 13.04 from live usb.  I know that in order to be able to keep files/settings/etc after restart, USB live disk should be created with some storage reserved. However since I got this USB from a friend who created installation, I don't know if he reserved any space, and so my questions are:   How can I find out amount of space reserved (if any)? If no space is reserved, can I manually chose USB/HDD location to store my settings in order to be able to boot into them the next time I log on?   I'm currently running a very long session and it would help me if I could somehow be able to store my settings (firefox, apt-gets, xrandr screen conf. ... ) to not have to do it again after I reboot.  Thanks,  IR Isomorphism between $\Bbb R$ and $\Bbb R(X)$? My questions are:     $1.$ Is there a field morphism $\Bbb R(X) \hookrightarrow \Bbb R$ ?      $2.$ If the answer to $1.$ is "yes", are $\Bbb R$ and $\Bbb R(X)$ isomorphic as fields?   $ $    $ $  For $1.$, here are some comments:   This is true for $\Bbb C$, see . The key point here is the fact that $\Bbb C$ is algebraically closed, of cardinality $2^{\aleph_0}$, and of characteristic $0$. The same argument doesn't work for $\Bbb R$. We can actually find a subfield $F$ of $\Bbb R$ such that $F \cong F(X) \cong F(X,Y) \cong \cdots$, see .  that $\Bbb Q(X)$ does not embed in $\Bbb Q$ (as a field).  This is equivalent to: Is there an injective ring morphism $\Bbb R[X] \hookrightarrow \Bbb R$ ? (By taking the fraction fields).   As for 2. :   They are isomorphic as abelian groups, and actually as $\Bbb Q$-vector spaces. They are not isomorphic as $\Bbb R$-algebras, because they haven't the same transcendence degree over $\Bbb R$. Let's try to see how to construct such a field isomorphism. We could try$^{[1]}$ to find a subset $B$ of $\Bbb R$ such that $\Bbb R = \Bbb Q(B)$ and $C \subsetneq B \implies \Bbb R \neq \Bbb Q(C)$.  The set $B$ must contain transcendental elements, pick such an $x_0 \in B$.  Then $\Bbb R = \Bbb Q(B) = \Bbb Q(B \setminus \{x_0\})(x_0)$ and we could try$^{[2]}$ to find a field isomorphism $\Bbb Q(B \setminus \{x_0\}) \cong \Bbb Q(B)$, so that $\Bbb Q(B \setminus \{x_0\})(x_0) \cong \Bbb Q(B)(x_0) \cong \Bbb R(X)$.  For ${[1]}$, I tried to use Zorn's lemma on $\mathscr B = \{B \subset \Bbb R \mid \Bbb R = \Bbb Q(B)\}$ and the partial order $B_1 ≤ B_2 \iff B_1 \supset B_2$. But this is not clear to me why this should be an inductive set.  For ${[2]}$, my idea was to define $f : \Bbb Q(B \setminus \{x_0\}) \cong \Bbb Q(B)$ by choosing a bijection $b : B \setminus \{x_0\} \to B$ (these sets should be uncountable), and then trying to extend $f\vert_{B \setminus \{x_0\}} = b$ to a field morphism.     Edit for $^{[1]}$: in  article (MINIMAL GENERATING SETS OF GROUPS, RINGS, AND FIELDS, by LORENZ HALBEISEN, MARTIN HAMILTON, AND PAVEL RUZICKA) the theorem 2.4. proves that $\Bbb R$ has no minimal generating set as field (or $\Bbb Q$-algebra, i.e. as ring). The argument uses Artin-Schreier theorem. Minimal generating sets of modules are   .  I think that $B$ is a minimal generating set of a subfield $K \subset \Bbb R$ over $\Bbb Q$ if and only if any element of $B$ cannot be written as a rational fraction of other elements of $B$. Under these conditions and using (transfinite) induction, we can probably prove (and this was my first idea) that any function $f: B \to \Bbb C$ such that $b$ and $f(b)$ are conjugate over $\Bbb Q$ (i.e. have the same minimal polynomial) extends to a (unique) field morphism $\sigma : K = \Bbb Q(B) \to \Bbb C$.  Actually, the collection $\mathscr B$ defined above doesn't have to be an inductive set, at least if we replace $\Bbb R$ by $K =\Bbb Q(\sqrt 2, \sqrt[4]{2},\sqrt[8]{2},\dots)$, because the descending chain $\left(E_n = \{\sqrt[2^m]{2} \mid m \geq n\}\right)_{n \geq 1}$ satisfy $K = \Bbb Q(E_n)$, but $K \neq \Bbb Q\left(\bigcap_{n≥1} E_n\right)=\Bbb Q$, so the chain has no upper bound w.r.t. $≤$ in $\mathscr B$. Word for a person being used I'm looking for a word to describe someone who is being used. I want a noun to describe this person, not a verb or or adjective.  Maybe like a pushover. I made the jump from Jumsoft's Money app to GnuCash a while back. It appealed to all my OCD tendencies. I don't know how to use Perl, or stuff of the sort, but with detailed instructions I manage to get things done... when I can find them.  I set up Finance::Quote to pull quotes after I set up GnuCash. I even managed to get it to pull quotes for Korean funds. And every thing had been working well, until tonight. I only get back a pop-up that states "There was an unknown error while retrieving the price quotes." That's where I need help. What can I do?  I dug around and found that it seems that it has something to do with Yahoo Finance ending it's API support. I also learned that Finance::Quote can pull data from many sources, so I don't really understand why one missing API would break it. I also saw that someone out there is trying to use AEX to pull the quotes instead for GnuCash specifically. I found Finance::Quote on cpan (I also first heard of cpan), but I don't understand what I'm looking at. But that's all beside the point. I don't know how to fix this.  Any help would be appreciated. I'm on Windows 10, GnuCash version 2.6.17, and have Strawberry Perl 5.18.2.2-32bit installed.  edit Thanks everyone who responded. It seems to be working more or less. The currency and Korean funds still haven't updated, but you've shown this horse to water. I'll do my best to drink it now. Override a method at instance level My teacher proposed this question as a challenge proof to do on our own and I can't seem to get it. Was Wondering if anyone could give me a hint or help me on the process to completing it.  Let $a \in \mathbb{R}$. Let $f$ be a function defined, at least, on an interval  centred at $a$, except possibly at $a$. Let $L \in \mathbb{R}$.     If $\lim\limits_{ x→a-} f(x) = \lim\limits_{ x→a+} f(x) = L$   then $\lim\limits_{ x→a} f(x) $ = $L$.    We are supposed to prove this using the "delta-epsilon" proof style. Thanks in advance!! My book uses the argument that the multiplicities of a few macrostates in a macroscopic object take up an extraordinarily large share of all possible microstates, such that even over the entire lifetime of the universe, if each microstate had an equal chance of being accessed, fluctuations away from equilibrium would never occur.  My question to this statistical proof is this: In the real world, is there really an infinitesimal but nonzero chance that macroscopic systems could access some of the more unlikely macrostates (e.g. heat transfer from a cold object to a hot object)? Why does locking the rear tires on a vehicle cause it to spin? are often used in motor-sports and stunt driving to turn around tight corners.  Essentially, the driver applies the handbrake as he enters the turn, in order to "kick out" the back end and slide the vehicle around the corner.  However, applying the handbrake hard enough to lock the rear wheels while traveling straight will cause the rear of the vehicle to "kick out".  Given enough speed, the vehicle will spin 180 degrees.  Why do sliding wheels on a car want to "get in front" of rolling wheels? I can write  VAR=$VAR1 VAR=${VAR1} VAR="$VAR1" VAR="${VAR1}"   the end result to me all seems about the same. Why should I write one or the other? are any of these not portable/POSIX? How can we prove that a binomial coefficient $n\choose m$is divisible by the ratio of $n$ and $\gcd(n,m)$? I am currently working to design a flash virtual tour, but I know little about how the photographs are taken.  Ideally I'd like to be able to take a set of 6 shots from a single focal point, and be able to align them seamlessly into a cube form. The other solution is to take a special fisheye image, and stretch it dynamically in flash.  I've seen examples of both as virtual tours, and ones formatted as a rotating cube are more responsive and have a higher image quality. The airbus and Cruden Homes are definitely in the cube format, I'm not entirely certain which format the New York VT uses:        I have a set of questions, the first is the most important, the others may become separate questions if I don't get an answer here:   How are those photos taken? Is there a specific mount that can be used? How long does it take to set up? How much post-processing is necessary on the images? I want to delete the record type which is default to the some profiles Record Type *Record Type Label    Sales Order *Record Type Name     Sales_Order  Namespace Prefix Description  Active        This record type Sales Order cannot be deactivated because the   following profiles use this record type as default.  This record type Sales Order cannot be deactivated because it is used by the following Objects.      Profiles- System Administrator, Standard User, CC_Customer Community   Plus Login Admin CC_Customer Community Plus Login User   CC_Marketing_Profile CC_Sales_Profile Chatter External User Chatter   Free User Chatter Moderator User Customer Community Plus Login User   Customer Community Plus User Deployment Manager Profile Read Only   Solution Manager Marketing User Contract Manager Pardot Sync User   Profile Objects CC_PO_Update Test_Utility   CC_CreateSalesOrderController_Test how can i show site logo, name and slogan Separately in custom area in my site theme, I now use this code to display theme, buts this code don't allow me to edit each of them Separately.  {{ page.header }} Iterating each character in a string using Python How to reset Kali linux root password Interpolating functions returned by NDSolve do not plot Why crontab scripts are not working? Often, crontab scripts are not executed on schedule or as expected. There are numerous reasons for that:   wrong crontab notation permissions problem environment variables   This community wiki aims to aggregate the top reasons for crontab scripts not being executed as expected. Write each reason in a separate answer.  Please include one reason per answer - details about why it's not executed - and fix(es) for that one reason.  Please write only cron-specific issues, e.g. commands that execute as expected from the shell but execute erroneously by cron.