Suppose $f:[a,b]\to\mathbb{R}$ is continuous. Prove that $$2\int_a^b\left[f(x)\int_x^bf(y)\,dy\right]dx=\left[\int_a^bf(x)\,dx\right]^2.$$   I got to $$\left[\int_a^bf(x)\,dx\right]^2=\int_a^bf(x)\left[\int_a^xf(y)\,dy+\int_x^bf(y)\,dy\right]dx$$ and I hoped to show that as $x$ ranges from $[a,b]$, the integrals $\int_a^x$ and $\int_x^b$ might take on the same values, but that's not the case. What should my next step be?
Suppose $f$ is a continuous single-variable function, prove that:  $$2\int_a^b \int_a^x f(x)f(y) \, dy \, dx = \left[ \int_a^b f(x) \, dx \right]^2$$  This question was just on my Calculus III final exam and I couldn't figure it out, could someone provide some insight, please?
How can I revert a config file back to the originally installed version after I have edited it? I've done a clean install of 18.04 and I can't open desktop folders, not even the desktop trash can. Anybody else had this issue? I click on the desktop folders (double click) and nothing happens.  Strange enough, I can open  them when using Nautilus and going to home\stackinator\desktop\ or whatever the path is (can't remember at this point). Should Suggested Edits change the technical content of a post? I have noticed a significant number of suggested edits that attempt to fix code, or change the original meaning of someone else's post.    Is this really a good idea?    I've been on StackOverflow awhile, and among the editors who have sufficient reputation to edit without approval, I've never seen anyone modify the technical content or meaning of someone else's post.   I would consider obvious syntax and spelling errors OK to change, but I have seen many suggested edits that change the meaning of the code that was posted, or add things to the code that have nothing to do with the original question. A good example of this would be adding parameters to a SQL command, when the question was not asking about SQL injection. A girl who transcends time I remember quite a bit, but can't seem to find the title or (female) author.   A girl accompanies her father as he captains a mission to travel at near-light speed in order to experience the effects of time dilation and return to Earth 100 years into its future. She leaves behind her boyfriend (Tal or Tel or something), who promises he'll find a way to stay with her despite the time travel that will render him, well, dead before her return. She's accompanied by a close friend who is deaf but lip-reads and speaks so well as to occasionally fool others into believing she is hearing. They spend a lot of time in the ship's garden. I believe he ends up either getting on a different ship to survive into the future via time dilation, or else having himself frozen. Either way, if I recall correctly, Earth isn't doing well at all by the time the mission with the girl returns. There's been a war or some other disaster. There's an extended period of vague, ambiguous, existential crap in which the girl (now a woman) eventually ends up transferring her consciousness into her own daughter, presumably in order to go on living. Everything wraps up somehow and the girl and her boyfriend walk together hand in hand on a desolate planet into the sunset near the end of time.   It was all really rather trashy and silly, I remember, and took itself entirely way too seriously, but I'd like to at least know what it was so I could read it again. Either it'll be worth a laugh, or it'll turn out to be surprisingly good. Each time I changed and saved .config/user-dirs.dirs, and after restarted the system, the XDG_DOWNLOAD_DIR are reverted:  diff --git a/.config/user-dirs.dirs b/.config/user-dirs.dirs index 8f6e2db..dcff34e 100644 --- a/.config/user-dirs.dirs +++ b/.config/user-dirs.dirs @@ -7,11 +7,9 @@  #   XDG_PUBLICSHARE_DIR="$HOME/public"  XDG_TEMPLATES_DIR="$HOME/public/template" -  XDG_DESKTOP_DIR="$HOME/private/desktop"  XDG_DOCUMENTS_DIR="$HOME/private/doc"  XDG_PICTURES_DIR="$HOME/private/picture"  XDG_MUSIC_DIR="$HOME/private/music"  XDG_VIDEOS_DIR="$HOME/private/video" - -XDG_DOWNLOAD_DIR="$HOME/var/downloads" +XDG_DOWNLOAD_DIR="$HOME/"   How to make my changes permanent? Prove that $B[x] \cap B[x^{-1}]$ is integral over $B$ Hole-and-nail paradox in special relativity Rotation of rigid body with two different angular velocities Consider a cylinder that rotates about a vertical fixed axis with angular velocity $\vec{\Omega}$ while rotating about  a vertical axis passing through its center of mass with angular velocity $\vec{\omega}$.    Now the two angular velocities of rotation $\omega$ and $\Omega$ are not the same I suppose and there is no link between the two angular velocities, is that right?  Firstly what is the angular velocity of the rigid body? I suppose that it is  $$\vec{\omega^{*}}=\vec{\omega}+\vec{\Omega}$$  But why?  And then what is the moment of inertia with respect to the axis $z$ of the rigid body? If I use parallel axis theorem I get  $$I_{z}=I_{cm}+mR^2$$  Where $R$ is the distance from $z$. But that doesn't seem right. The cylinder is not rotating about its center of mass at the same angular velocity of rotation about $z$. So using $(1)$ give some incorrect results. For istance kinetic energy  $$K=\frac{1}{2} I_{z} \vec{\Omega}^2= \frac{1}{2}(I_{cm}+mR^2)\vec{\Omega}^2$$  But that's the kinetic energy of a body rotating about $z$ and center of mass with the same angular velocity $\vec{\Omega}$, and this is not the case.  I'm confused, can anyone help me with give suggestion about angular velocity and moment of inertia in problems where the rigid body is rotating at the same time about two different axes? I'm exploring the effects of removing the intercept in a logistic regression model.  Assume a model:  $$logit(Y = 1) = \beta_1 x + \beta_2z + 0$$  with $x$ and $z$ being categorical variables with 2 levels each and no intercept.  I understood that having no intercept with categorical predictors produce coefficients that compare the $P(Y = 1)$ in each level of the two predictor against a null case where $P(Y=1) = 0.5$ or $logit(Y=1) = 0$.  I noticed a phenomenon that can understand. Using glm() function in R if you change the order of the variable in the right hand part of the formula, the coefficients change too. But even more oddly, the coefficient of the first variable is always the same.  Here's an R demo:  y &lt;- as.factor(sample(rep(1:2), 30, T)) x &lt;- as.factor(sample(rep(1:2), 30, T)) z &lt;- as.factor(sample(rep(1:2), 30, T))  coef(glm(y ~ x + z - 1, binomial) #        x1         x2         z2  #-0.1764783  0.3260739 -0.1335192  coef(glm(y ~ z + x - 1, binomial)) #        z1         z2         x2  #-0.1764783 -0.3099976  0.5025523    As you can see the first predictors have the same coefficient while the other are different in the two models.  Here is what I expected and instead behave differently than what I though:   Since every level of the two predictors is compared to the same null case, I expected to have the same coefficients in the two models, independently from the order in which I use them. I expected to see the coefficients of every level of every predictor, instead the coefficient for the 1 level of the second predictor is not shown. I therefore assume that only the first variable is compared against the null case, while the second is compared against a reference level; but what is this level? Is it $P(Y = 1 | X = 1 \cap Z = 1)$? Reproducing one of the models WITH the intercept we get:  coef(glm(y ~ x + z - 1, binomial) #        x1         x2         z2  #-0.1764783  0.3260739 -0.1335192  coef(glm(y ~ x + z, binomial)) #(Intercept)         x2          z2  #-0.1764783   0.5025523  -0.1335192    As expected x1 become the intercept, and x2 is likely relative to x1. z1 is missing also in this case and z2 is the same as in the model without intercept.  Thus should I assume that the comparison against the null case $P(Y = 1) = 0.5$ is made only for the first variable in a formula, while the other are compared against the usual intercept? Is this behavior normal? What about the fact that the first coefficient has the same value whichever the order of the predictors in the formula? What if I want to compare all level of each predictor against the null case and have a coefficient for all levels? Or it's theoretically impossible for some reason I don't get? How to show the full path of a file or directory in the terminal? I need to know how the directory name in order to type it out in the terminal. How do I access the names of directories?   Windows Explorer used to have a title bar with the full path. Can someone please help me figure out how to see the full path of a certain file? Finding the distribution of the sum of three independent uniform random variables How can I convert the binary string $x_bin="0001001100101" to its numeric value $x_num=613 in Perl? ColdFusion adding extra quotes when constructing database queries in strings I am coding in ColdFusion, but trying to stay in cfscript, so I have a function that allows me to pass in a query to run it with   &lt;cfquery blah >   #query# &lt;/cfquery>   Somehow though, when I construct my queries with sql = "SELECT * FROM a WHERE b='#c#'"  and pass it in, ColdFusion has replaced the single quotes with 2 single quotes.  so it becomes  WHERE b=''c'' in the final query.  I have tried creating the strings a lot of different ways, but I cannot get it to leave just one quote.  Even doing a string replace has no effect.   Any idea why this is happening?  It is ruining my hopes of living in cfscript for the duration of this project Why does a cordless circular saw have the blade on the left? At the moment considering the purchase of a circular saw for light work. I'm looking at a corded model and a cordless model. But the question I have is not about which one is better.   My question today is why the corded one has the blade on the right of my hand and the cordless one has the blade on the left of my hand. See both models below.     Now, I know some manufacturers make left- and right-handed versions of their circular saw. And I'm also aware that 'worm drive circular saws' usually have the blade to the left of my hand. But it seems odd to me. Especially as it seems to apply to most manufacturers too.   Was wondering if somebody has an explanation for this. Prove that for real number $p\geq  0$ we have   $$\lim_{n\to\infty}\frac{(1^{1^p} \cdot2^{2^p}\cdots n^{n^p})^\frac{1}{n^{p+1}}}{n^\frac{1}{(p+1)}} = \exp\left(\dfrac{-1}{(p+1)^2}\right).$$ Is there a difference between "write" and "write down"? Displaying layout marks in a document set with different layouts Is there any package that allows for the display of layout marks within a document where the layout changes several times throughout? I'm using the geometry facilities to issuing \newgeometry commands throughout my document, but it doesn't update the displayed layout and neither does the showframe package or the facilities of the memoir class.  EDIT: As Werner has pointed out, the showframe option of the geometry package displays correctly any updated page layout set by issuing \newgeometry commands.   My trouble came out of the fact that I'm issuing also \pagecolor commands and the page color seems to be hiding the layout lines typeset according to the geometry package. My confusion only worsened because the showframe package layout lines were not hidden by the page color (the package got the layout wrong on some pages, though). So, it all boils down to a conflict with the page color.   Even though I'm using geometry's showframe option for "debugging" ends, and I don't mind to have no page color at that stage of development, is it possible to override this problem and have at the same time both the page layout displayed by geometry as well as the page colored?   Btw, should I open a new question given that the problem is actually different from the originally posted one? What do you call a person who is always online on the Internet? Is there any specific word for a person who is always online on the Internet?  I am just curious to know because staying online is like a profession nowadays. Which rationals can be written as the sum of two rational squares? Which rational numbers can be written as the sum of two rational squares?  That is, for which rational numbers $a$, are there rational numbers $x$ and $y$ such that $a = x^2 + y^2$.  It is a famous theorem that if an integer can be written as the sum of two rational squares then it can be written as the sum of two integral squares, and then the solution is the famous one by Fermat, but I didn't find anything about the general case. As the photo - Variable 'n' has already been declared.  I've tried clearing my cache, as well as using a new user profile with no extensions, but it doesn't work.  Might be useful to know so it can be fixed before more people are using Chrome (If it's SO's fault of course!)