I know that by integrating the surface area of a sphere we can get the formula of its volume.  Similarly, I wanted to find about other ways of doing this. So, I thought what if we divide the sphere into infinite circles or sections of the sphere and the add their areas for the volume and their perimeter for the surface area.  Problem is I am not able to do this. I tried but didn't work for me. I just knew the above logic but didn't know how to implement it.  So, I would like you to guide me on how I can do this and not just give a direct solution. Also, I would like to know about other ways for getting these results.   P.S.- if you are just giving me the answer can you explain it a bit. My calculus is not very strong. I know basic rules but I don't know how to set up the equations and I think that's where I am weak. How to implement this kind of logic.
Why does the integral of the surface area of a sphere equal it's volume?   $$\int{4\pi r^2 \ \mathrm{d}r =\frac{4}{3}\pi r^3}$$  I don't quite understand why the relationship between surface area and volume is this way. Is this true for any other shapes?
How can I make schematic of this board? Is it possible to view the scene in the viewport camera view at 1:1? How do defenders of libertarian freewill reconcile it with constraints imposed by the laws of physics? Floor function properties: $[2x] = [x] + [ x + \frac12 ]$ and $[nx] = \sum_{k = 0}^{n - 1} [ x + \frac{k}{n} ] $ I'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one:     DEFINITION Given  $x\in \Bbb R$, the integer part of $x$ is the unique $z\in \Bbb Z$ such that $$z\leq x &lt; z+1$$ and we denote it by $[x]$.   Now he asks to prove some basic things about it, such as: if $n\in \Bbb Z$, then $[x+n]=[x]+n$  So I proved it like this: Let $z=[x+n]$ and $z'=[x]$. Then we have that  $$z\leq x+n&lt;z+1$$  $$z'\leq x&lt;z'+1$$  Then $$z'+n\leq x+n&lt;z'+n+1$$  But since $z'$ is an integer, so is $z'+n$. Since $z$ is unique, it must be that $z'+n=z$.  However, this doesn't seem to get me anywhere to prove that  $$\left[ {2x} \right] = \left[ x \right] + \left[ {x + \frac{1}{2}} \right]$$  in and in general that   $$\left[ {nx} \right] = \sum\limits_{k = 0}^{n - 1} {\left[ {x + \frac{k}{n}} \right]} $$  Obviously one could do an informal proof thinking about "the carries", but that's not the idea, let alone how tedious it would be. Maybe there is some easier or clearer characterization of $[x]$ in terms of $x$ to work this out.  Another property is $$[-x]=\begin{cases}-[x]\text{ ; if }x\in \Bbb Z \cr-[x]-1 \text{ ; otherwise}\end{cases}$$  I argue: if $x\in\Bbb Z$, it is clear $[x]=x$. Then $-[x]=-x$, and $-[x]\in \Bbb Z$ so $[-[x]]=-[x]=[-x]$. For the other, I guess one could say:  $$n \leqslant x &lt; n + 1 \Rightarrow  - n - 1 &lt; x \leqslant -n$$  and since $x$ is not an integer, this should be the same as $$ - n - 1 \leqslant -x &lt; -n$$  $$ - n - 1 \leqslant -x &lt; (-n-1)+1$$  So $[-x]=-[x]-1$ Why does Calculate button remain gray in Group Stats Plugin for QGIS? In my case I failed to use  plugin (2.0.30) with QGIS 2.14.11 and 2.18.3 on windows64.  Tested on several datasets: points, polygons, and csv.- But I am not sure what I have missed.    I want to click on this Calculate button. Get all categories and posts in those categories I am looking for a solution that allows me to print the following:  Cat 1        Cat 2        Cat 3  Post 1       Post 1       Post 1  Post 2       Post 2       Post 2  Post 3                    Post 3                            Post 4   EDIT  I am looking for something that will only require one database query! So if you have a foreach in your code followed by a new WP_Query then that is not what I am looking for (I am planning to put this onto my website's homepage). I have a table similar to this:  CREATE TABLE example (   id integer primary key,   name char(200),   parentid integer,   value integer);   I can use the parentid field to arrange data into a tree structure.  Now here's the bit I can't work out. Given a parentid, is it possible to write an SQL statement to add up all the value fields under that parentid and recurse down the branch of the tree ?  UPDATE: I'm using posgreSQL so the fancy MS-SQL features are not available to me. In any case, I'd like this to be treated as a generic SQL question.  BTW, I'm very impressed to have 6 answers within 15 minutes of asking the question! Go stack overflow! How to deal with perfect separation in logistic regression? If you have a variable which perfectly separates zeroes and ones in target variable, R will yield the following "perfect or quasi perfect separation" warning message:  Warning message: glm.fit: fitted probabilities numerically 0 or 1 occurred    We still get the model but the coefficient estimates are inflated.   How do you deal with this in practice? I read with interest about Einstein's Theory of Relativity and his proposition about the speed of light being the universal speed limit.    So, if I were to travel in a spacecraft at (practically) the speed of light, would I freeze and stop moving?  Would the universe around me freeze and stop moving?  Who would the time stop for? When typing &lt; or &gt; in LaTeX and compiling with pdflatex, the less than and greater than symbols appear at upside down exclamation points. I'm not in math mode. Let be the next equivalence relation on $B= \{R \subseteq \mathbb{N}\times\mathbb{N} \ | \ R \ is \ a \ well-order \ on \  \mathbb{N}\}$ : $$"R \equiv R' \leftrightarrow \ (\mathbb{N},R) \cong (\mathbb{N},R') \ are \ well-ordering \ sets \  isomorphic" $$  Which is the cardinality of $B/\equiv$?  Firstly, we can notice that each well-ordering set is isomorphic to an ordinal $\alpha \ $such that $\alpha &lt; \aleph_{1}$, and so it is enough to count how many ordinals are isomorphic to a subset of $\mathbb{N}$. For example, if we consider $A$ the set of the even numbers and $B$ the set of odd numbers with the usual order relation, $A \oplus B$ is isomorphic to $\omega + \omega$. Another example could be: if we consider the order of $\mathbb{N}$ that we use in Sharkowski Theorem, that order is isomorphic to $ \omega^{2} + \omega$. Working with prime numbers we could build also $\omega^{n} \ \forall n \in \mathbb{N}$, and so $\omega^{\omega}$ and maybe something bigger. However, we know that each countable ordinal is isomorphic to a subset of $\mathbb{Q}$, but I bet that the same result is not true for $\mathbb{N}$ because it is "too small".  What can I do? Including pdf_tex images from inkscape $\pi$ can be represented as $C/D$, and $C/D$ is a fraction, and the definition of an irrational number is that it cannot be represented as a fraction.   Then why is $\pi$ an irrational number? Many anime do not follow the manga history and are made shorter. However there are a few that do follow it or try to finish the history with the manga like the big three (Naruto, Bleach, One Piece), Death Note, and FMA Brotherhood. But others finish the anime without the real ending like FMA, Ouran High School, Pandora Hearts, Tsubasa Reservoir Chronicle. Why is this so? Is it because of the high cost? Or the lack of popularity? What's the difference between a temp table and table variable in SQL Server? "Unable to connect to Upstart" error when installing git-all I am using 15.04.   So I'm working with git and I installed the standard version. I set up the ssh keys and was able to clone one of my repositories to the correct folder. Everything was working great and then I decided it was a good idea to install more packages. I ran sudo apt-get install git-all. I'm looking at error codes now having to deal with upstart.  start: Unable to connect to Upstart: Failed to connect to socket /com/ubuntu/upstart: Connection refused No apport report written because the error message indicates its a followup error from a previous failure. dpkg: error processing package runit (--configure):  subprocess installed post-installation script returned error exit status 1 dpkg: dependency problems prevent configuration of git-daemon-run:  git-daemon-run depends on runit; however:   Package runit is not configured yet.  dpkg: error processing package git-daemon-run (--configure):  dependency problems - leaving unconfigured Errors were encountered while processing:  runit  git-daemon-run E: Sub-process /usr/bin/dpkg returned an error code (1)   This is what happens when I even update/upgrade my system. At this point all I'm looking to do is be able to install software and update my current software. Is there any way I can potentially reverse this issue or at least correct it?   Edit: Thankfully this issue was solved only a day or two after it happened. The accepted answer worked perfectly and took care of my issue Prove that $\lambda = 0$ is an eigenvalue if and only if A is singular. Apollonius circle, its radius and center How to make a film in black and white and add color to some objects? How is a movie made to show selective colors. For instance, in  a sister and brother are transported to the universe of an old black and white TV show, and the movie switches to being B&amp;W. Then, as the protagonists cause the inhabitants to have 'real' emotions, like fright, anger, or love, things (people, places, etc.) start having color. In the middle of the movie, many objects are still B&amp;W, yet many more are now colorized.   also used this effect (the whole film was initially shot in full color, and was converted to black-and-white. Colorization is used on certain subjects in a scene), for example.  How is this effect achieved? Is it post-production editing? What techniques might a director employ to do this? Can I get a postdoc in China more than 3 years after having received my PhD?