Recent content by Parthalan

I *think* I get it now. Because 4x^2+9y^2=36 doesn't extend above y=2, I needed to "partition" the volume and evaluate \pi \left ( \frac{5}{4}\int_0^2{x^2\,dx} + \int_2^3{9-x^2\,dx} \right ) = 6\pi and then either double for the part below the x-axis, or work it out by hand using the same...

Homework Statement The area enclosed between the ellipse 4x^2 + 9y^2 = 36 and its auxiliary circle x^2 + y^2 = 9 is rotated about the y-axis through \pi radians. Find, by integration, the volume generated. This is the whole question. I assume it means bounded by the x-axis, but even if...

Hi, I need a little help with my understanding of g-forces. From what I can gather, this "force" is the acceleration experienced by the object expressed as a multiple of g (the usual acceleration due to gravity). What has confused me, though, is the Wikipedia page, which says that "A...

In the notation f:A \rightarrow B, A is the domain, and B is the codomain, of which the range (which is \{y \in B : y = f(x), x \in A\}) is a subset.

If you're looking for something less computational, you might want to look into Finite-Dimensional Vector Spaces by Paul Halmos, considered by many to be one of the greatest expositors in mathematics of his time. If you're after a reference, then look elsewhere, but it is a beautiful book for a...

I happen to have the 7th edition of Brown and Churchill, and it has my recommendations. It's well written and clear, and it will have the information you need in it. From what I can gather of Bak and Newman (from Amazon's Search Inside), it's talky, and a bit more gentle. In addition to...

Or did you mean a piecewise function, with: f(x) = \left\{ \begin{matrix} x^2, & \mbox{if } x \leq a\\ x^3, & \mbox{if } x > a \end{matrix} I don't know if functions defined like this are considered elementary, but in this case, f:\mathbb{R} \rightarrow \mathbb{R}. I'm not sure from your...

Any excuse to use hyperspherical coordinates (I think) will do!

It doesn't have an inverse in the typical sense of a function, however, it can be thought of as a multivalued function, and these have been studied extensively. Consider the example of x\mapstox^2. Then its inverse is \pm \sqrt{x}. If we choose by convention to take the positive roots, we have...

I think the point Gib Z is making (correct me if I'm wrong) is that, since \tan^{-1}(x) is multivalued, we're using the branch cut of (-\frac{\pi}{2}, \frac{\pi}{2}). This is our choice of principal values, and then the limit is indeed \lim_{x \rightarrow \infty} \tan^{-1}(x) = \frac{\pi}{2}.

Well the general solution we discovered involved polylogarithms, and apparently \mathrm{Li}_s(1) = \zeta(s) where \Re(s) > 1, but that still doesn't really answer the question! I can't even tell what I was thinking. Sorry.

If you have to do it with algebra, have you thought about using substitution with u = \tan^{-1}(x) \implies du = \frac{1}{1+x^2}\,dx to get \int \ln(1+x^2) \cdot \left [ \tan^{-1}(x) \right]^2\cdot u\,du?

I managed to get \frac{3}{64}[7\pi^2\zeta(3) - 31\zeta(5)] from Mathematica, though you might want to double check that.

I think the way to show that it can't be integrated in terms of an elementary function is using the Risch Algorithm (see http://mathworld.wolfram.com/RischAlgorithm.html" ). As for what the algorithm actually is, I have no idea!

The row rank of a matrix is the number of linearly independent rows, and similarly, the column rank is the number of linearly independent columns. Most importantly, note that the two are always equal! With A = \left [ \begin{array}{cccc} 2 & 4 & 1 & 3 \\ -1 & -2 & 1 & 0 \\ 0 & 0 & 2 & 2 \\ 3 &...