Help with evaluating a double integral How would I go about evaluating...  $$\int_{0}^{2}\int_{0}^{4-x^2} \frac{xcos(3y)}{4-y}dydx$$  I don't really have any work to show, since I don't even know how to start this problem, but I'll offer my thought process to show you what I've thought about and we'll see where to go from there.  I can't think of any particular reason why circular, spherical or cylindrical substitutions would make sense here. For this reason, I feel like a different type of substitution would be necessary...but...what? I don't know what type of substitution would make this easier. I couldn't just say that $ u = 4-y $, because in the numerator, I already have that nasty $ cos(3y) $ term.   Any help would be greatly appreciated.
Double integral of an function odd with respect to $y$ over a domain symmetric with respect to $x$ axis Let $D$ be a region given as the set of $(x, y)$ with $$a \leq x \leq b\quad\text{and}\quad-\Phi(x) \leq y \leq \Phi(x)$$ where $Φ$ is a nonnegative continuous function on the interval $[a, b]$.  Let $f(x, y)$ be a function on $D$ such that $$f(x, y) = - f(x, - y)$$ for all $(x , y) \in D$.  Argue that $$\displaystyle \iint_D f(x, y) dA = 0.$$
Left ideals of $M_n(K)$ If you have the ability to cast a spell without a spell slot, can you cast that spell using a spell slot? A 10th level Drow warlock (for instance) has the following Drow Magic racial trait (PHB p. 24), in addition to its warlock spellcasting abilities:  You know the dancing lights cantrip. When you reach 3rd level, you can cast the faerie fire spell once per day. When you reach 5th level, you can also cast the darkness spell once per day. Charisma is your spellcasting ability for these spells.  Can the Drow warlock cast faerie fire (which is not on the warlock spell list) using a warlock spell slot? Can it cast darkness (which is on the warlock spell list) using a warlock spell slot?  The  says this about the Magic Initiate feat:  If you’re a spellcaster, can you pick your own class when you gain the Magic Initiate feat? Yes, the feat doesn’t say you can’t. For example, if you’re a wizard and gain the Magic Initiate feat, you can choose wizard and thereby learn two more wizard cantrips and another 1st-level wizard spell. If you have spell slots, can you use them to cast the 1st-level spell you learn with the Magic Initiate feat? Yes, but only if the class you pick for the feat is one of your classes. For example, if you pick sorcerer and you are a sorcerer, the Spellcasting feature for that class tells you that you can use your spell slots to cast the sorcerer spells you know, so you can use your spell slots to cast the 1st-level sorcerer spell you learn from Magic Initiate. Similarly, if you are a wizard and pick that class for the feat, you learn a 1st-level wizard spell, which you could add to your spellbook and subsequently prepare.  But this specifically requires you to pick the feat for one of your classes to be able to cast the spell learnt using your class spell slots, and does not seem directly applicable. A Warlock can use warlock Pact Magic spell slots to cast spells learnt from levels of another spellcasting class, per the multiclass spellcasting rules (PHB p. 164):  If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and you can use the spell slots you gain from the Spellcasting class feature to cast warlock spells you know.  Similarly, if the warlock has learnt the invocation Mask of Many Faces (PHB p. 112)...  You can cast disguise self at will, without expending a spell slot.  ...Can the warlock cast disguise self using a spell slot, e.g. to decrease the chance it can be dispelled? Is it possible to legally visit uninhabited islands in the Pacific for a couple of days/weeks like Robinson Crusoe Do you know any way how can I visit uninhabited islands in the Pacific (French Polynesia, Micronesia etc.) and stay there for a couple of days or weeks?  The only condition, these island (or islands) should have drinking water.  Could it be done legally or only illegally?  Does any company do such crazy things via plane or boat?  Any information is appreciated.  I am mostly interested in French Polynesia.  Boat or plane... that's not very important. How to recover deleted files? Windows 8.1 64-bit: High memory usage I have 8GB of memory on a 64 bit windows 8.1 machine, and windows is telling me I'm using 91% of my memory, yet its simultaneously telling me I'm only using 3GB.  I literally just restarted my machine a few minutes ago because it was being incredibly rediculously slow. I'm currently only running chrome (with a ton of tabs, but doesn't add up to anywhere near 7.5GB). What could be going on here?   Here's a bunch of my current info       I took a look at this:  but I wasn's able to use xperf and the rest didn't lead me to a solution.  UPDATE - RamMap:   UPDATE 2: I checked Resource Monitor at Wouter's suggestion and found an "Image" called lxdecoms.exe with over 6GB of Commt (KB). I tried killing the process, but while it lowered my Commit Charge, it didn't affect "Used Physical Memory". So I restarted hoping to capture it again in full, but its not back yet and yet the problem persists.  I should note that I do have a Centos virtual box virtual machine running on my machine, not sure if that has any weird memory impacts that aren't reflected in the numbers in these tools.   I got xperf working, so here's my resource monitor and a wpa analysis of my xperf output (from xperf -on ReferenceSet -BufferSize 1024 -MaxFile 512 -FileMode Circular  &amp;&amp; timeout 5 &amp;&amp; xperf -d MemUsage.etl):     What is N/A that's taking up 2.7GB of memory? My VM? Blender 2.78 with Windows 10 lags when interacting with the UI How do I hide the GRUB menu showing up at the beginning of boot? I stopped dual booting and I don't want to have to choose the OS any more. I want to boot into Ubuntu 12.04 directly without seeing the GRUB menu.  How can I achieve that? What and how does Darth Vader see? Are there any sources that talk about  about Darth Vader's visual acuity?  His life-supporting armour is quite advanced but does it provide any useful augmentation to his visual abilities? Use a baseline including all observations in lm() How do I rip audio from a DVD? Formatting section titles Forgive the stupid simplicity of this question, and please feel free to refer me to an older answer. I'm not quite familiar enough with LaTeX terminology to have found it.  All I want to do is set a different default format for my section titles. Instead of just being bolded and slightly larger than the main text, I'd like them to be of normal weight, the same size as the main text, and all caps.   I'm using the article class. Proving that expression is equivalent to the definition of derivative Let $f$ be differentiable at $x=a$.   Prove that if $x_n \to a^+$ and $y_n \to a^-$ then:  $$\lim_{n\to \infty} \frac{f(x_n)-f(y_n)}{x_n-y_n}=f'(a).$$  Every option that I think about seems to my very trivial, so I believe that I am doing something wrong.  Both numerator and denominator approach zero as $n\to\infty$ as the case of the formal definition of derivative, but it isn't guaranteed that the limits are equal (“$\frac{0}{0}$”).   Any direction? Converting ArcGIS (*.mxd) file to QGIS (*.qgs) file? How to change permissions of multiple files found with find command? Do airlines have to check the visa status before selling a ticket? What can a wizard take instead of a familiar at 1st level? The familiar is brilliant feature for a wizard, but not for a prestige multiclassing one. Such a wizard stops upgrading his familiar once getting off the main road. So if he still wants to utilize his familiar, he should take the obtain familiar feat, but this would made the basic familiar feature redundant.  So what options can a wizard take instead of gaining a familiar at 1st level? What does (obj.length === +obj.length) compare? Meta duplicates don't require an answer. Don't suggest there is one when that's not true This request was sparked . The vote to close as a duplicate on meta, in contrast to the main site, does not require for the duplicate to have an answer. Yet, the closure message still says:     This question has been asked before and already has an answer.    ...even when this is not the case. Perhaps we should either only show this particular sentence if there is indeed an answer, or remove this altogether for meta.   Note: I fear this might be a duplicate request, but I was unable to find one. (Not even one without an answer :) ) How $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2 = \frac{\sum_{i=1}^n (X_i - \bar X)^2}{n}$ How   $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2 = \frac{\sum_{i=1}^n (X_i - \bar X)^2}{n}$   i have tried to do that by the following procedure:  $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2$  =$\frac{1}{n}(\sum_{i=1}^n X_i^2 - n\bar X^2)$  =$\frac{1}{n}(\sum_{i=1}^n X_i^2 - \sum_{i=1}^n\bar X^2)$  =$\frac{1}{n} \sum_{i=1}^n (X_i^2 - \bar X^2)$  Then i have stumbled. Ubuntu 18.04 can't resume after hibernate Hibernate worked correctly when I had been using Ubuntu 17.04. After upgrade to 17.10 I couldn't resume because of black screen after resume ().   After upgrade to 18.04 if I try to start my computer after hibernation it boots as if there was no hibernation done.