The question tells me to find the volume of a spherical cap with height $h$ and radius $r$ (by radius I mean radius of the cap, not the whole sphere).  Well I attempted to solve it, as $R$ is the radius of the sphere, $r =\sqrt{R^2-y^2}$ via Pythagorean theorem. Now $A$ is the cross-section of the sphere, which is $\pi\big(\sqrt{R^2-y^2}\big)^2$. That can be simplified as $\pi(R^2-y^2)$.  Then I integrated it in terms of $R$ and $R-h$ (the height of the spherical cap), and got $\pi(h^2R-h^3/3)$. But how do I express it in terms of $h$ and $r$, where $h &lt; r$?
A solid spherical cap of radius $R$ is the solid delimited by a sphere    of radius $R$ and a plane that intersects the sphere.   The height $h$ of the solid spherical cap is the largest distance from a point of the solid to the plane intersecting the sphere.   Show that the volume of a solid spherical cap is given by:  $$V = \frac{\pi h^2(3R-h)}{3}$$   It is enough to consider the solid spherical cap delimited by the sphere $x^2 + y^2 + z^2 = R^2$ and the plane $z = R-h$  use triple integrals by spherical or cylindrical coordinates. Give me a few min to try it out.
Can anyone suggest a detailed installation process for Ubuntu on Windows 8.2? Classify all groups with order $pq$, $p$ is not equal to $q$ I am reading the following material:  . At page 5, there is a proof classifying all groups with order $pq$(they are all primes and $p$ is not equal to $q$). I am lost in the last two sentence, which claims the semidirect product are all isomorphic. Could someone help me to understand it? Thanks! Should an apostrophe be used with a z at the end of a word? I know that if a word ends with an s then it should be made plural or possessive by use of an apostrophe with no extra s (e.g. "Thomas Harris' book was a riveting read." as opposed to "Gary Barlow's voice is so distracting.")  What about if the word ends with a z. For example, would it be Gomez' album or Gomez's album? deploying and retrieving Address settings (State & Country picklists) Distance from a point to a plane in normed spaces How can we calculate the distance from a point to a plane in normed spaces ? where we are not inner product, for example:  Calculate the distance in $(C[0,1],||\cdot||)$ endowed with the supremum norm $$||f||=\sup_{t\in[0,1]}|f(t)|$$ from the function $f(t)=t^2$ to the linear hull of the functions $g(t)=t$  and  $h(t)=\sin(t)$  Any hints would be appreciated. Best way to specify whitespace in a String.Split operation How do I extract an ISO on Linux without root access I want to put a link to my SO profile on my homepage. I remember a codesnippet i saw somewhere that would basically display my Username and my reputation. Where was that? How can it be that the beginning universe had a high temperature and a low entropy at the same time? The Big Bang theory assumes that our universe started from a very/infinitely dense and extremely/infinitely hot state. But on the other side, it is often claimed that our universe must have been started in a state with very low or even zero entropy.   Now the third law of thermodynamic states that if the entropy of a system approaches a minimum, it's temperature approaches absolut zero.  So how can it be that the beginning universe had a high temperature and a low entropy at the same time? Wouldn't such a state be in contradiction to the third law of thermodynamics? Redirect output from all commands to file, not just last I have a 1TB FAILING hard drive, a brand new 1 TB hard drive, a bootable ubuntu usb, and a working computer.  All I need is a step by step process of how to safely clone as much data as possible from one to the other. NOTE: THE DRIVE IS FAILING.  I know I need to enable multiverse and then install and use ddrescue to complete the task. But everywhere I look online, I can't get a straight path to the task. I know that this can be all done within the terminal and that's what I seek.  How can I do this? Seeking podcasts related to geospatial technologies? I would like to make a list of podcasts related to geospatial technologies.  Do you listen to any that you find interesting and informative? Can I run a natural gas dryer from a portable propane tank? Why do people buy stocks that pay no dividend? How can I get a reference to a method? Is there a dominant gene for right-handedness? Has there been any definitive research about handedness being genetic? Also, why is right-handedness clearly dominant in humans? I'm interested in evolutionary theories, as well as any molecular explanation (if known). Show that $A$ is an invertible matrix if $\left(A+I\right)^3=0$ and find $A^{-1}$ How to get keycodes for xmodmap? I'm trying to use xmodmap to remap Alt / Super keys on Dell L100 keyboard, and have trouble getting the keycodes.  For instance, using xev doesn't give me keycode for Alt   FocusOut event, serial 36, synthetic NO, window 0x4a00001,     mode NotifyGrab, detail NotifyAncestor  FocusIn event, serial 36, synthetic NO, window 0x4a00001,     mode NotifyUngrab, detail NotifyAncestor  KeymapNotify event, serial 36, synthetic NO, window 0x0,     keys:  122 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0               0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0      For Right Super key, xev and showkey give different keycodes -- 134 and 126 respectively.  What's going on with these keycodes?  I tried getting keycodes from showkey -k, and using xmodmap file below, but that gave a weird map which remapped b key:  clear Mod1 clear Control keycode 125 = Meta_L keycode 126 = Meta_R keycode 58 = Control_L keycode 56 = Control_L keycode 100 = Control_R add Control = Control_L Control_R add Mod1 = Meta_L Meta_R I was showing a GM friend of mine a build I was making that uses the Two-Handed Fighter Archetype and we are both unsure of how it interacts with the damage bonus you get from two-handed weapons (which is 1-1/2 times your STR.) It reads:     Overhand Chop (Ex): At 3rd level, when a two-handed fighter makes a single attack (with the attack action or a charge) with a two-handed weapon, he adds double his Strength bonus on damage rolls. This ability replaces armor training 1.   So, does Overhand Chop turn the 1-1/2 times multiplier from two-handed weapons into a 2 times multiplier, or do the two add up? Or, do they interact in some other way?   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