All aspects of integration, including the definition of the integral and computing indefinite integrals (antiderivatives).
13
votes
0answers
639 views
Computing ${\int\limits_{0}^{\chi}\int\limits_{0}^{\chi}\int\limits_{0}^{2\pi}\sqrt{1- \cdots} \, d\phi \, d\theta_1 \, d\theta_2}$?
This question led me to this integral I can not solve:
$$
...
10
votes
0answers
144 views
definite and indefinite sums and integrals
It just occurred to me that I tend to think of integrals primarily as indefinite integrals and sums primarily as definite sums. That is, when I see a definite integral, my first approach at solving it ...
8
votes
0answers
289 views
a way of evaluating integrals without doing anything?
The user known as sos440 posted this:
$$\begin{align*} \sum_{n=0}^\infty \frac{r^n}{n!} \int_0^\infty x^n e^{-x} \; dx & = \int_{0}^\infty \sum_{n=0}^\infty \frac{(rx)^n}{n!} e^{-x} \; dx = ...
7
votes
0answers
307 views
Cauchy-Formula for Repeated Lebesgue-Integration
Recently, I came across the following statements. They were annotated as consequences of Fubini's Theorem but neither proof nor reference were given.
Let $f:[a,b]\times [a,b]\to\mathbb{R}$ be ...
6
votes
0answers
56 views
Asymptotic Expansion of an Oscillating Integral
Let $g(x):\mathbb{R}_{\geq0}\rightarrow\mathbb{R}$ be real analytic s.t. $g(0)\neq 0$ and $g(x)=O(x^{-2})$ as $x\rightarrow\infty$.
What is the leading order in $\lambda$ as $\lambda\rightarrow 0$ of ...
6
votes
0answers
148 views
Computing the volume of a region on the unit $n$-sphere
I would like to compute the surface volume of a region on the unit $n-1$-sphere:
$$x_1^2 + \dots + x_i^2 + \dots + x_n^2 = 1,$$
bounded by an ellipsoid
$$a_1x_1^2 + \dots + a_ix_i^2 + \dots + ...
5
votes
0answers
62 views
What is the relation of $\int f dx^1\wedge dx^2\wedge …\wedge dx^n=\int f dx^1…dx^n$
In a book "calculus on manifolds" it is defined that $\int f dx^1\wedge dx^2\wedge ...\wedge dx^n=\int f dx^1...dx^n$ but how it is possible the relate the integrand of a multilinear function ...
5
votes
0answers
220 views
Is this question solvable? $2$ non-linear equations and the proof that the solution is unique (with asymmetric bounty option)
As mentioned in the title I want to show the uniqueness of the solution to $2$ non-linear equations. However, it seems that I can not solve this question with my current mathematical knowledge. More ...
5
votes
0answers
96 views
Nontrivial trivial integrals
Consider two propositions in geometry:
Circumscribe a right circular cylinder about a sphere. The surface area of the cylinder between any two planes orthogonal to the cylinder's axis equals the ...
5
votes
0answers
123 views
An integration to first order
I am having some trouble evaluating an integral -- involving taking an approximation. It would be great if someone could help me.
I wish to evaluate
$$\int_0^\pi {\cos\theta\cos \left[\omega ...
5
votes
0answers
162 views
Evaluating $\int{ \frac{\arctan\sqrt{n^{2}-1}}{\sqrt{n^{2}+n}}} dn$
How to integrate?
$$\int{ \frac{\arctan\sqrt{n^{2}-1}}{\sqrt{n^{2}+n}}} dn$$
I have no idea how to do it.
Tried to get some information from wiki, but its too hard :|
5
votes
0answers
51 views
Various integration theories
Could anyone briefly explain, or point me towards a resource explaining, the main differences between the main integration theories, namely:
Riemann Integration
Riemann-Stieltjes Integration
...
5
votes
0answers
188 views
To determine if$f^{-1}(x)$ is periodic function or not? $f(x)=\int_1^{x} \frac{1}{\sqrt[m]{P(t)}}\;dt$
$$f(x)=\int_1^{x} \frac{1}{\sqrt[m]{P(t)}}\;dt$$
$P(x)$ is polynomial with degree $n$.
$m$ is an positive integer and $m>1$
What is the algoritm to determine $f^{-1}(x)$ is periodic function ...
5
votes
0answers
114 views
Evaluting $ \int_0^{\infty}\frac{v}{\sqrt{v + c}}e^{-\frac{y^2}{2(v + c)} - \frac{(u-v)^2}{u^2v}}dv$
While working on mixture (variance) of normal distribution and keep running into these two integrals
$$ \int_0^{\infty}\dfrac{v}{\sqrt{v + c}}e^{-\dfrac{y^2}{2(v + c)} - \dfrac{(u-v)^2}{u^2v}}dv,$$
...
4
votes
0answers
47 views
Integration by parts of a normalized function - [copied from Physics.SE]
By using integration by parts, I need to show for $$A = \frac{\mathrm d}{\mathrm dx} + \tanh x, \qquad A^{\dagger} = - \frac{\mathrm d}{\mathrm dx} + \tanh x,$$ that
...
