I've solved quite a bit of these questions with tetrahedrons with a 90 degree angle, but I am not sure how to approach a non-right angle tetrahedron. I am unsure how to find the triple integral's bounds. Calculation itself is no problem.   $ \int_B (x + y + 2z) dV$  where B is the tetrahedron with vertices $(0, 0, 0)$, $(0, 2, 0)$, $(1, 3, 1)$ and $(1, 1, 0)$.  Is the way of finding the bounds same for non-right angle tetrahedrons?
Find the volume of the tetrahedron with the vertices $P(1,1,1)$, $Q(1, 2, 3)$, $R(3, 1, 2)$, and $S(2, 3, 1)$.
Bayes' Theorem Intuition I've been trying to develop an intuition based understanding of Bayes' theorem in terms of the prior, posterior, likelihood and marginal probability. For that I use the following equation: $$P(B|A) = \frac{P(A|B)P(B)}{P(A)}$$ where $A$ represents a hypothesis or belief and $B$ represents data or evidence. I've understood the concept of the posterior - it's a unifying entity that combines the prior belief and the likelihood of an event. What I don't understand is what does the likelihood signify? And why is the marginal probability in the denominator? After reviewing a couple of resources I came across this quote:     The likelihood is the weight of event $B$ given by the occurrence of $A$ ... $P(B|A)$ is the posterior probability of event $B$ , given that event $A$ has occurred.   The above 2 statements seem identical to me, just written in different ways. Can anyone please explain the difference between the two? I've noticed something while trying to prove the properties of parallel lines, and the properties that a triangle has 180 degrees. To prove the properties of parallel lines, such as alternate angles, you need to use the property that a triangle has 180 degrees. To prove a triangle has 180 degrees however, you need to use the properties of parallel lines. This really bothers me because of how circular it is. They are both reliant on each other to be true, and do not logically show, without being reliant on each other, why triangles have 180 degrees, and why parallel line properties are true.   So what I hoping for here is a way to prove parallel line properties without using the fact that a triangle has 180 degrees, or a way to prove triangles have 180 degrees without using parallel line properties. This way, things will be logical to me, and make sense.   Thanks in advance. Can't install MySQL on Mountain Lion Query works on Oracle 11g but fails on Oracle 8i Is "this Monday" or "next Monday" the correct way to refer to the very next Monday in the future? Possible Duplicate:          When I refer to the very next Monday that will occur in the future, I say "next Monday". Some colleagues refer to it as "this Monday", with "next Monday" meaning the second Monday which will occur in the future (I would refer to that as "Monday week", "this Monday" to me would mean the most recent Monday in the past).  Are these both acceptable usages, or is one more correct than the other? System Load Indicator only shows as a line unless hovered I installed the  GNOME extension on my Ubuntu 18.04, but for some reason, instead of showing a proper graph for CPU usage, it only shows a very thin line that seems to be a small portion of the actual graph. If I hover over it, the proper graph shows. How to make it show the whole graph all the time? I made up some integrals to do for fun, and I had a real problem with this one. I've since found out that there's no solution in terms of elementary functions, but when I attempt to integrate it, I end up with infinite values. Could somebody point out where I go wrong?  So, I'm trying to determine: $$ \int{\frac{e^x}{x}} \, dx $$  Integrate by parts, where $u = 1/x$, and $v \, ' = e^x$. Then $u \, ' = - 1/x^2$, and $v=e^x$. So,  $$\int{\frac{e^x}{x}} \, dx = \frac{e^x}{x} + \int{\frac{e^x}{x^2}} \, dx$$  Integrate by parts again, $u = 1/x^2$, $v \, ' = e^x$, so that $u \, ' = -2/x^3$ and $v=e^x$. So,  $$\int{\frac{e^x}{x}} \, dx = \frac{e^x}{x} + \frac{e^x}{x^2} + 2\int{\frac{e^x}{x^3}} \, dx$$  Repeat this process ad infinitum to get,  $$\int{\frac{e^x}{x}} \, dx = \frac{e^x}{x} + \frac{e^x}{x^2} + 2 \left( \frac{e^x}{x^3} + 3 \left( \frac{e^x}{x^4} + 4 \left( \frac{e^x}{x^5} + \, \cdots \right) \right) \right) $$  Expanding this gives,  $$\int{\frac{e^x}{x}} \, dx = \frac{e^x}{x} + \frac{e^x}{x^2} + \frac{2e^x}{x^3} + \frac{6 e^x}{x^4} + \frac{24 e^x}{x^5} + \cdots $$  And factoring that gives,  $$\int{\frac{e^x}{x}} \, dx = \frac{e^x}{x} \left(  1 + \frac{1}{x} + \frac{2}{x^2} + \frac{6}{x^3} + \frac{24}{x^4} + \cdots \right) $$  Now, considering the series itself, the ratio between the $n^{th}$ term and the $(n-1)^{th}$ term = $\Large \frac{n}{x}$. Eventually, $n$ will be larger than $x$, so the ratio between successive terms will be positive, so (assuming $x$ is positive), the series diverges, meaning (and I'm sure everybody will cringe upon seeing notation used like this), that:  $$\int{\frac{e^x}{x}} \, dx = \frac{e^x}{x} \left( \infty \right) = \infty $$ I have a big pot full of chicken necks and backs on the stove which I need to separate into (a) stock and (b) everything else.  I'm wondering the best way to go about doing this. I've previously laid cheesecloth into a fine-mesh strainer and poured it through that, but the cheesecloth always seems to kind of clog up, making straining 4 litres of stock a very lengthy process. I also tried a coffee filter laid inside a strainer and got more-or-less the same results.   Alternatively, I ditched the cheesecloth and just used the fine-mesh strainer by itself, but, unsurprisingly, this method leaves some bits I'd rather not have.   I'm wondering: is there a better way? Or should I stick with the cheesecloth and work on being more patient? Unity smooth local rotation around one Axis, oriented toward a target Identifying a quotient group (NBHM-$2014$) Let $\mathbb C^*$ denote the multiplicative group of non-zero complex numbers and let $P$ denote the subgroup of positive real numbers. Identify the quotient group.    My thought $$\frac{\mathbb C^*}{P}=\{P,-P,iP,-iP\}.$$ Is it right? Install Windows 7 with Bootcamp on OS X Yosemite with USB Key I have an iMac (bought in 2011) and I recently upgraded it with Yosemite. I am trying to install Windows 7 on it via Bootcamp.  However, its optical drive does not work, so I chose the USB key solution using the trick of modifying the file info.plist of Bootcamp (type 'usb bootcamp youtube' in google to see a video of this trick).  So I follow the process religiously and end up with the classic and famous boring screen 'No bootable device -- insert boot disk and press any key'.  So I tried to use Unetbootin (  ) to make my USB Key bootable, and tried to boot on it through the screen that appears when you maintain the Option key (Alt button) while starting the mac. Result: I never managed to have my Unetbootin-formatted USB key appear on this screen :(  I read some stuff about reFit (  ) and GParted, but I just don't know how to use them...  If someone managed to install Windows 7 on his/her iMac with Yosemite using a USB Key, please help me understand why I can't see the USB as a boot option. Displaying world country shapefiles centered on Pacific Ocean using Robinson or Miller Cylindrical projection in QGIS? I wish to display a map in QGIS (world country shapefiles) showing all countries but centered on the Pacific area.   I am not familiar with Proj4, so is there any way this can be done in QGIS? I am looking for a novel that I'd love to re-read it with my wife (my 13-year old self must be proud of me I guess). Key Plot points The plot was about humanity finding an alien station on an orbit of Earth. (I'm not sure if it wasn't Moon itself). On this station there were thousands of ships ready to travel so there rose a company that paid people to get on those ships and explore the galaxy. The problem was humans didn't understand how to steer those ships so they had to rely on preset targets. Of course there was no telling where particular ship was to go and fair amount of them never returned. Chronology and other relevant information I read it as an early teenager back in early 2000s. It was Polish translation of a foreign (possibly English) story. I think (not sure) that the book wasn't new at the time. It was just one of those books you dig out in a library and take to see if it's good. I don't recall cover either and even if I did it'd be of the Polish version. Additional information could be that the narration was led along the psychoanalytics of the main protagonists where he relives his memories from inside the station and his journeys. I've been Googling around with no luck.  I'm looking for a way to do getc() or gets() or whatever that does not echo to the terminal.  I saw kbhit() but this doesn't appear to be part of ANSI.  Ideally, I'd like code that looks like  char s[100]; no_echo_gets(s); /* Won't echo to screen while being typed */ printf("%s\n", s);   Does anybody know a good ANSI compliant way to do this? Cannot get 2 raster files of same cell size to align ArcMap 10.2 What's the difference between using single and double quotation marks/inverted commas? My supervisor asked me to find out which distribution represents a particular situation.  I have a VoIP generator that generates calls "uniformly" distributed between callers. This means that the volume per caller distribution is "almost" uniformly distributed between a minimum and maximum. So by running a test with 10000 users and a min value equal to 30 calls per week and a max value equal to 90 calls/week i obtain that not all the users respect this limits: we have some users that generate &lt;30 calls and some other users that generate >90 calls. It is clear that the obtained distribution is not uniform.  The situation is this:    He said that i have to perform a sort of numerical process in order to find some formulas that could define this distribution. Initially, as wrote before,we wanted to obtain a uniform (min,max) distribution (the green area in the figure) but this is not the case as proved with chi-square test.  Moreover the curve in figure is not symmetric, the probability of call generation below 30 call/week and greater that 90 call/week are not identical (it is high for 90calls/week). The variability of the number of generated call increases with the increasing call generation rates.   "Actually implementation of this distribution is nothing but assigning different call rates in a range for users in domain which indicates implementation of several delta functions. As the call rates increases the variability of the generated calls also increases with the average call rate and this leads to the asymmetric behavior of the curve." [cit. from the Voipgenerator documentation]  Someone can help me?I think that now i cannot use Q-Q plot because i don't know which theoretical distribution i have to use in order to compare it with my empirical data.  Sorry if I have stressed with a similar problem a few weeks ago, but initially we thought we could change the implementation, but now we cannot. Hence i have to discover the type of the distribution i obtained and i don't know how can i do this. How can we prove that a binomial coefficient $n\choose m$is divisible by the ratio of $n$ and $\gcd(n,m)$? What is the "continue" keyword and how does it work in Java? I saw this keyword for the first time and I was wondering if someone could explain to me what it does.   What is the continue keyword? How does it work? When is it used? Is "there're" (similar to "there's") a correct contraction? Q: "Do you have any juice?"   A: "Yes, there's some in the fridge."   Sounds perfectly fine to me, but:     Q: "Do you have any towels?"   A: "Yes, there's some in the closet."   Does not.  I asked for towels - plural - so wouldn't "Yes, there're some in the closet," in which there are is turned into a contraction be the correct way to say it?  Spellcheck, however, doesn't like "there're", and I think I'm the only person I've ever heard use the word "there're".  Even folks who I know say "there are" shorten it to "there's" when possible.  Am I saying it wrong, or are both forms acceptable?