Dave

Try rewriting the equation using a = sin(x), b = cos(x) and factorising it.

At my girlfriend's house atm, dialup == suckage

Your integral is right, you will get the volume of a sphere out, but you need to just work at it a bit more. I went through the entire thing for you, and I'll provide the proof if you really want it.

I think a change of co-ordinates into spherical co-ordinates may be the ones you're looking for, not cylindrical. The triple integration then just becomes trivial.

I like fish.

Most of the modern calculators do these days and I think I've got one of those in my draw anyway. I daresay Atlantic just put it into his calculator wrong.

Liked the one where they first met the Borg. Very good episode.

I'd kindof disagree there. The problem with C++ is the fact that because it is such a powerful language, it means that syntaxically, the language is very complex. For a first time programmer, you don't want to be in a position where you don't understand a piece of code that you've just written (as I found myself doing quite a lot of times). I started out a very long time ago using basic (the old BBC version) then moved onto QBasic, Visual Basic, TCL (scripting language they use for IRC bots and other things as well), C, and then eventually C++. I'd wholeheartedly recommend VB as a first language to get your head around the entire programming mindset and solving simple algorithms, getting you introduced to simple data types, etc. The only problem with this is that you'll find that when you do make the transition from VB to C/C++ (btw: recommend you do C before you start moving onto C++) is that since you can program things in VB with fairly complex graphical interfaces in just a couple pages of code, you can't really do this in C++. It requires quite a bit more effort, but it's definately worth it because you can literally do anything you put your mind to with a language as powerful as C++. Another language I'd definately recommend just to introduce you to object orientated programming (which is implemented in C++ quite a lot) is Java. Again it's not as hard to create GUI's as in C++, but it is a very good language to introduce you to the basic concepts. The other big advantage to learning Java instead of VB first is the fact that syntaxically, C++ and Java are very similar - in fact, there's hardly any differences at all in most cases. Hope this helps.

This was a great source of confusion for my maths class a few years back when we were learning the basics. Say if you have [math]x^2-4=0[/math]. Then [math]x^2=4[/math]. So you want to find a value for [math]x[/math] such that when you square it, it decides to become 4. However, when you draw the graph of the function, it becomes fairly clear that there's actually 2 solutions. The problem is that when you square a negative number, the answer is positive. So in this case, the answer would involve a plus/minus sign. -- I noticed this a while back, all the math tags actually do is encapsulate the stuff you put in them and execute a cgi script from the looks of it. You can't really have spaces in the request you send to the webserver easily so I think mimeTeX just ignores them.

Currently on Warwick University LAN, which has a rather large pipe. Get speeds of about 1.5mb/sec (off of decent servers anyway).

Rolle's theorem can still be applied, but you'd consider it on the function f:[d,e]->R (assuming of course that d < e).

probably just a question mark, I think the Ti-83 has some kind of text editor capability.

If you want overflow, just get it to work out 1000! Not even my TI-89 can handle that (I don't think anyway)

fingers

To be fair, aerodynamics of the coin have very little role to play considering the forces involved; i mean, yes, if you had a 50m coin, and tossed it in a 70mph gale, then I think we'd take it into account, but a 2cm coin, launched into the air at a not very high speed and in a moderately windless environment will encounter negligable forces due to the air resistance and aerodynamical properties of the coin.

Just in the middle of reading my Vector Calculus book (yes, I have an assignment due in tomorrow and I haven't done it yet, would there be any other reason for me reading this insanely stupid book?)

The classic example for positive binomial is items (for some reason it's usually light bulbs) that come off of a production line. i.e., the probability of a light bulb being faulty is 0.3 (assume this independent and whatnot). if you have a random batch of 20, what's the probability that exactly 12 of them are not faulty? Call the number of non-faulty light bulbs X => X ~ B(20, 0.7) P(X=12) = nCr(20, 12)*(0.7)^12*(0.3)^8 = whatever. Hope this helps.

Use the laws of indices: [math]\(a^{b}\)^{c}=a^{bc}[/math]. So [math]a^{m/n}=\(a^{m}\)^{1/n}=\(a^{1/n}\)^{m}[/math]

Think about the meaning of 'square root'. If you try and find the square root or some number n, it's another number p such that p^2 = n. For n = 4, there exist two such numbers: -2 and +2. It's all a matter of definition.

For non-negative values of x, the graphing problem is fairly trivial. It's when you extend the domain of the function to include negative numbers that you run into problems. When x is a negative integer, then the function is defined, since you can have something like (-2)^(-2) = 1/4. It's when you have x belonging to the set of rationals that you run into quite a few problems, since it'll work for some values and not for others. For instance, take x to be of the form (2p/q) where p is a negative integer and q is a positive integer (obviously with p and q not equal to zero). So you know that the numerator is always going to be an even number, so it'll involve making your negative x positive (effectively because you're squaring it p times). But when you have x as some number (p/q) where p is odd and q is even (since you can cube/fifth/seventh.. root a negative number) you're going to be effectively square rooting a negative number, which is obviously not defined in the real line. There are a lot of problems when x is any negative real number. Mathematically speaking, we can say that for all x in the open interval (-infinity, 0), x is discontinuous - it has lots of gaps in it, effectively making it impossible to graph properly. It also means it's not differentiable in this domain. But for the half-open interval [0, infinity), it's continuous - i.e. a smooth line. Of course, you can get rid of the problem. Let's say the function f:N->R (N is the set of natural numbers, R is the set of reals, and f maps some number x from the naturals to the reals). Then f(x) = x^x is defined for all x in the domain. If you continue with mathematics at university, you'll do a lot of this stuff. Hope this helps some.

I'm not entirely sure whether logarithmic functions were defined first, or whether they were looking for a function that, when the derivative was taken, gave the original function as itself (as in e^x). But anyway, it turns out that for things like radioactive decay, cooling curves, etc that the logarithmic function and e have a large role to play. For example, take your bog standard radioative decay equation: dB/dt = -:lclambda: t where :lclambda: is the decay constant. (not entirely sure about the B - may be Q instead?) to solve this equation, you need to do seperation of variables, which basically boils it down to this equation: :int: 1/(-:lclambda: t) = :int: dB and from the definition of the logarithmic function, the solution is something of the form: Q = Q0e^(-:lclambda: t) This equation crops up in a lot of places, but i'm not sure whether it crops up in your particular case.

It's stupid. The amount of information that satellite has provided is just phenominal. It seems completely stupid to stop maintaining it just because of some unreasoned political decision.

no

It can. However I thought this was a nice proof I'll see if I can find/derive the series proof for you at some point. Double integrals aren't all that hard really, it's just the same as normal integration more or less.

Thought I'd post this for your enjoyment. I thought it was quite nice. Enjoy. dblint.pdf