Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

learn more… | top users | synonyms (3)

2
votes
0answers
32 views

Integral of addition of measurable functions

Let $(X, \mathcal{M},\mu)$ be a measure space. Let $f,g: X \to \mathbb{R}$ (extended real line) be measurable functions. Prove that if both $\int f^{+} d\mu$ and $\int g^{+} d\mu$ are finite, or ...
2
votes
0answers
84 views

Show $f$ if integrable on $[a,b]$ if and only if $\epsilon > 0$, there exists a partition $P_{\epsilon}$ of [a,b]

Let $f$ be bounded on a nondegenerate interval $[a,b]$. Prove that $f$ if integrable on $[a,b]$ if and only if $\epsilon > 0$, there exists a partition $P_{\epsilon}$ of [a,b] such that P is a ...
2
votes
0answers
35 views

How can we compute the integral of a Laplacian of a radial function over an open ball

Let $B_R\subseteq\mathbb{R}^n$ be an open ball with radius $R>0$ centered at $0$ and $f\in C^0\left(\overline{B_R}\to\mathbb{R}\right)$ be a radial function, i.e. $f(x)=f(r)$ with ...
2
votes
0answers
44 views

Spherical Cardioid

Calculate the mass of the apple-shaped solid bounded by the rotated cardioid $\rho = R(1 - \cos \varphi )$ (in spherical coordinates), if the density at distance $\rho $ from the origin is given ...
2
votes
0answers
34 views

What are the X,Y coordinates of round beads strung along an Archimedian spiral of string?

I have a commercial application for which I have simplified the underlying mathematical problem to the following: There are 32 spherical beads on a string, each of diameter 'd'. Each bead is touching ...
2
votes
0answers
26 views

Center of mass and circular paraboloid

The solid $W$ below is bounded by the circular paraboloid $$z = 2a\left( {1 - \frac{{{x^2} + {y^2}}}{{{{(3a)}^2}}}} \right)$$ and the $xy$ plane. At the point $(x,y,z) \in W$ its density is $$\delta ...
2
votes
0answers
32 views

Integrating a function of measures

I've been reading John Baez's series of posts on Information Geometry. I'm currently on part 6... Midway through the post he discusses Radon-Nikodym derivatives: The formula for information gain ...
2
votes
0answers
58 views

Is there a closed-form expression for this trigonometric Cauchy Principal Value-type integral?

Consider the following definite integral, $I(n; \theta)$. $$ I(n; \theta) = \int_{0}^{\pi} \frac{\cos(n\phi)}{\cos\phi-\cos\theta} d\phi \quad \text{where } n \in N $$ When $0 < \theta < \pi ...
2
votes
0answers
39 views

Dyson-expansion like multidimensional integral

Let $n \ge 1$ be an integer. Now let $0 \le t_0 \le t$ and $\beta \neq 1$ be real numbers. Now, let $\vec{p} := (p_0,p_1,\cdots,p_n)$ be strictly positive integers. Also let $(x)_{(n)} := x(x-1)\cdot ...
2
votes
0answers
120 views

Finding the volume of a cube using spherical coordinates

Calculate the volume of a cube having edge length $a$ by integrating in spherical coordinates. Suppose that the cube have all the edges on the positive semi-axis. Let us divide it by the plane passing ...
2
votes
0answers
40 views

The set composed of domain and codomain of integrable function measure zero

There is this problem which I have constructed a plan to prove, and I am stuck. If anyone could see my plan and tell what is wrong about it I would be very thankful. Let $f: Q \to [0,1]$ be ...
2
votes
0answers
153 views

How to evaluate this integral$\int_{-\infty}^\infty\dfrac{\omega^\alpha e^{i\omega t}}{(\omega_0^2-\omega^2)^{2}+4(\zeta\omega_0\omega)^2}\,d\omega$

How to calculate the following integral? $$\int_{-\infty}^\infty\dfrac{\omega^\alpha e^{i\omega t}}{(\omega_0^2-\omega^2)^{2}+4(\zeta\omega_0\omega)^2}\,d\omega$$ where ...
2
votes
0answers
41 views

Find an integrating factor such that $y'=\frac{1-x+y}{x-y}$ is exact

Yet another question of this sort, and hopefully the last. In the previous question I posted, we were lucky enough and the integrating factor was a function of only one variable, the ansatz $\mu_y=0$ ...
2
votes
0answers
65 views

Is this limit finite?

What is the limit of $$\lim_{u+v\rightarrow 1}\frac{\ln \int f_0(y)^{1-v} f_1(y)^{v}\mathrm{d}y- \ln \int f_0(y)^u f_1(y)^{1-u}\mathrm{d}y}{1-(u+v)}$$ where $f_0$ and $f_1$ are some density ...
2
votes
0answers
27 views

Riemann Integrals and Periodic Functions

Consider the following result, where integrals are, say, Riemann integrals: Let $f:\mathbb{R}\to\mathbb{C}$ be a periodic function of period $L$. Then for all $a\in\mathbb{R}$ $$ ...
2
votes
0answers
94 views

2 logarithmic integrals twins

This question also has the value of an answer to the first integral here How to evaluate $\int_0^1\frac{\ln(x)\ln(1+x^2)}{1-x}dx$ and $\int_0^1\frac{\ln(x)\ln(1+x^2)}{1+x}dx$ ...
2
votes
0answers
39 views

Computation of double integral $(1-xy)^b$

I want to determine the values of $b>0$ such that $$\int_0^1\int_0^1(1-xy)^{-b}dydx$$ exists and is finite. I think that the integral is finite for $0<b<2$ and infinite for $b\geq 2$ but I am ...
2
votes
0answers
32 views

Euler-Maclaurin formula for half integer values in summation

I am trying to use the Euler-Maclaurin formula to approximate a sum with the form $$\sum_{n=0}^\infty f(n+1/2)$$ where the argument is a half integer. Can anyone help me adjust the formula for such a ...
2
votes
0answers
45 views

The best student

Suppose two students called A and B. Student A has answered $k_A$ questions correctly out of $n_A$ questions. Student B has answered $k_B$ correctly out of $n_B$ questions. Who is the best student ...
2
votes
0answers
54 views

Can the minimum be given by an integral?

for $a,b > 0$, $$ \begin{align} &\int_{0}^{\infty} \frac{\sin (ax) \sin (bx)}{x^{2}} \ dx \\ &= \int_{0}^{\infty} \frac{a \cos (ax) \sin (bx) + b \sin(ax) \cos(bx)}{x} \ dx \\ &= ...
2
votes
0answers
31 views

Sum of a series of integrals of increasing order

I have a series and I am wondering under what conditions of $g(t)$ does it converge, and what does it converge to? $$ \lambda \int_{0}^{x}g(t) \, dt + \lambda^2 \int_{0}^{x} \int_{0}^{t} g(t_1) \, ...
2
votes
0answers
191 views

Prove generalized fundamental theorem of calculus

I need to prove the following generalized version of the FTC but I'm unsure how this is even different to the 'non-generalized' FTC. Let $F:[a,b]\to \mathbb R$ be continuous and piecewise ...
2
votes
0answers
66 views

Lebesgue-Stieltjes integral w.r.t. measure defined by absoluting continuous $F$

I know that if $F:[a,b]\to\mathbb{R}$ is a non-decreasing absolutely continuous function then$$\int_a^b f(x)dF(x)=\int_a^b f(x)F'(x)d\mu$$where the first integral is the Lebesgue-Stieltjes integral ...
2
votes
0answers
78 views

Is there a function whose definite integrals are all 0?

Is there a continuous function $f: [0,1] \rightarrow \mathbb{R}$ such that $f(x) \neq 0$ for some $x \in [0,1]$ and, if we define $F_n(x) = \int_{0} ^ {x} F_{n-1}(t) dt $ (where $F_0(x)=f(x)$), then ...
2
votes
0answers
34 views

Fluid Flow: lubrication, integration, ODE

Basically, I'm modelling the flow of a "coating" process -- a fluid flow between a flat moving plane and a stationary cylinder, 2D, cartesian coordinates. Subscript 0 is the at the minimum height b/w ...
2
votes
0answers
565 views

Relation I found: $ (\sum_{r=1}^{\infty}\frac{z(r)}{r})\times \int_0^\infty f(x) dx = \lim_{h \rightarrow 0} \sum_{i=0}^{n} f(k_ih)h$

I was fiddling with some maths and came up with an interesting relationship: $$ (\sum_{r=1}^{\infty}\frac{z(r)}{r})\times \int_0^\infty f(x) dx = \lim_{h \rightarrow 0} \sum_{i=1}^{n} f(k_ih)h$$ ...
2
votes
0answers
52 views

Calculating the Integral of a non conservative vector field

I have no clue how to do part C because a) is non conservative What I got for b) $f(x,y)=\frac{x^3}{3}+2yx+\frac{y^3}{3}+K$ (I don't know the symbol for the thing so I used f(x,y) instead. How do I ...
2
votes
0answers
39 views

How to relate two integration contour?

How one can relate two integration contour? For example if I have an integration contour like $\int_{-a}^{a}f(x)dx$ here let say a=infinity. How I can say that the integral $\int_{2}^{3}f(x)dx$ is a ...
2
votes
0answers
104 views

Multivariable integral over a simplex

Let $p$ be a positive integer, let $B > A >0$ and let $\beta >0 $ and $\beta \neq 1$. With a help of Mathematica (ie using elementary integration and consecutive simplifications) I have shown ...
2
votes
0answers
27 views

If $f$ is increasing, then for all $n\in\mathbb{N}$ there exists $P_n$ : $U(f,P)-L(f,P) \leq (b-a)/n$

I've already proven that, if $f:[a,b] \to \mathbb{R}$ is continuous and increasing, with $a,b\in \mathbb{R}$, then $$U(f,P) - L(f,P) = \sum_{i=1}^{n}\left[ f(x_i) - f(x_{i-1})\right](x_i - x_{i-1})$$ ...
2
votes
0answers
51 views

Time to buy a house without a mortgage equation!!

I am looking into a "real world" calculation to calculate the time taken for someone to buy their own home while they rent it. They do this by buying small pieces of the property every month, and ...
2
votes
0answers
84 views

Definite integral similar to beta function but with exponential negative square root

I'm trying to solve the following definite integral: $\mathcal{I} = \int_0^1dx\ x^{P+k/2-m}(1-x)^me^{-\sqrt{x}}, $ where $P\in\mathcal{N}$ (whole positive numbers and zero), $m\in\mathcal{N}$, ...
2
votes
0answers
67 views

Integral with cosines and power or upper bound

I need to solve this integral or find its upper bound $$\int_M^\infty \frac{2}{(t^2 \pi^2 + \epsilon^2)^\beta}\sin(\pi x t)\sin(\pi z t)\mathrm{d}t$$ I got to simplify upstairs as $$\int_M^\infty ...
2
votes
0answers
302 views

Convolution of two Gaussians or two sinc functions using direct integration

I tried to solve the following to problems from Gaskil's book Linear Systems, Fourier Transforms, and Optics. But I'm struggling to get the right results. My experience with calculating convolutions ...
2
votes
0answers
171 views

Interesting problem of finding surface area of part of a sphere.

Show that the surface area of a zone of a sphere that lies between two parallel planes is $2\pi Rh$, Where $R$ is the radius of the sphere and $h$ is the distance between the planes. If you are ...
2
votes
0answers
39 views

Is it possible to abstract a Riemann integral into a “higher” integral with measure?

I'm not very comfortable with more generalised integrals such as the Lebesgue integral yet, but I'm working through some material to achieve that goal. I have a question which stems simply from ...
2
votes
0answers
48 views

An upper Bound for $(f(a))^2$, $a\in[0,1]$ in terms of $\int_0^1(f(x))^2dx$

Is there any way to find an upper Bound for $(f(a))^2$, $a\in[0,1]$ in terms of $\int_0^1(f(x))^2dx$. There is a commonly used upper bound in terms of $\int_0^1(f_x(x))^2dx$, but I do want to make ...
2
votes
0answers
35 views

Skew symmetric matrices even size commutativity

Given Data in the question $w(t)=\frac{1}{2}\begin{bmatrix} 0 &r(t) &-q(t) &p(t) \\ -r(t)& 0 &p(t) &q(t) \\ q(t)& -p(t) &0 & r(t)\\ -p(t)&-q(t) ...
2
votes
0answers
34 views

Differentiation - a technical point

I understand the following equation to be correct, but why can we treat the differentials as fractions and cancel them out? What would be the correct way to view it? $$ \int_{-\pi/a} ...
2
votes
0answers
25 views

Generalized change of variables in integral

When I read the following (http://www.math.helsinki.fi/~analysis/GraduateSchool/maly/gs.pdf ), it is hard to understand it. In particular, what does it mean by the last equation? Why does it make ...
2
votes
0answers
26 views

Proof that maximal interval of existence exist and bounded

For each $\lambda\in \mathbb{R}$, let $\varphi_{\lambda}$ : $J_{\lambda}\rightarrow \mathbb{R}$ denote the solution to the following initial value problem: $$ ...
2
votes
0answers
44 views

Triple integral - volume of solid described by inequalities

I have to calculate the volume of solid described by inequalities: $$(x\leqslant y)\vee (y\leqslant z) \vee (x\leqslant z)$$ in region $[0,1]^3$. What is important, here we have conjunction. It is ...
2
votes
0answers
51 views

Multiple integral involving exponential and trigonometric functions

By using the generating function for Bessel functions I have discovered the following identity: \begin{eqnarray} &&\int\limits_{[0,2 \pi] \times [-\frac{\pi}{2},\frac{\pi}{2} ]^2} e^{\imath x ...
2
votes
0answers
152 views

Wikipedia proof of “Darboux Integral implies Riemann Integral”

I am trying to follow the proof at the page http://en.wikipedia.org/wiki/Riemann_integral which shows how a Darboux Integrable function is Riemann Integrable also. In general a partition of an ...
2
votes
0answers
32 views

Expressing indefinite integrals in terms of a predefined set of functions.

It is well known that some integrals of elementary functions cannot be expressed as elementary functions. I was wondering if it was possible to extend the set of elementary operators by some ...
2
votes
0answers
51 views

Simple Integral Involving the Square of the Elliptic Integral

I have, $$ \int uE^{2}\left(u\right)du $$ where $E$ is the complete elliptic integral of the second kind: $$ E\left(k\right)=\int_{0}^{\frac{\pi}{2}}d\theta\sqrt{1-k^{2}\sin^{2}\left(\theta\right)} ...
2
votes
0answers
32 views

Normal Vector Affecting The Divergence Theorem

$\newcommand{\Div}{\operatorname{Div}}$I'm going to use an example to explain what I'm trying to ask. Let $T =\{(x,y,z): x^2+y^2=z^2, 0\leq z\leq3\}$, I'm asked to calculate $\iint_T ...
2
votes
0answers
106 views

Definite Integral involving matrices

We have a definite integral of the form given below $ f(t) = \int_0^1 e^{\alpha X(t)} \frac{dX(t)}{dt} e^{(1-\alpha) X(t)}\,d\alpha \tag 1$ Given Data in the question $X(t)$ is a ...
2
votes
0answers
66 views

Can this modified Gaussian integral be calculated analytically?

In my research, I encounter this modified Gaussian integral $$\int_{-\infty}^{\infty}dx\,\frac{x+\sqrt{x^2-bx}}{2\sqrt{x^2-bx}}\exp\left[-a^2(x-x_0)^2+i\left(cx-d\sqrt{x^2-bx}\right)\right],$$ where ...
2
votes
0answers
68 views

Homotopy, Stokes Theorem and Orientation

I have a problem in which the theory and the computation disagree about a minus sign. My question requires a little setting up. I have a complex valued 2-form $$ \omega = \alpha(\xi_1,\xi_2)\, ...