All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

learn more… | top users | synonyms (2)

2
votes
0answers
368 views

Approximation of integral using series expansion of the integrand.

I have a smooth function $x \rightarrow f_\epsilon (x)$ on $x\in[-1\ldots 1]$ (dependent on the continuous parameter $\epsilon$) and I want to approximate the integral $$ I=\int_{-1}^1 f_\epsilon ...
2
votes
0answers
153 views

Normalization constant

This is probably a very obvious, but I am slightly confused. Suppose $f(\xi)$ is such that when $f^2(\xi)$ is integrated from $\xi=-\infty $ to $\xi=+\infty$ equals to $1$ and correspondingly for ...
2
votes
0answers
180 views

Closed form of the series ,$\sum_{k=0}^{\infty}\frac{(-1)^k k!}{(k+1)^{(k+1)}}x^k$

$x,y>0$ $$f(x,y)=\int_{0}^{\infty} \frac{1}{xt+e^{y t}} dt$$ if $x=0$ then $f(0,y)=1/y$ $$f(x,y)=\int_{0}^{\infty} \frac{1}{e^{y t}(1+xte^{-y t})} dt=\int_{0}^{\infty} \frac{e^{-y ...
2
votes
0answers
124 views

Find the height of the centre of mass of an annular hemisphere

So as stated above I need to find the centre of mass of a annular hemisphere. the outer radius is $R$ and the inner radius is $a$. In a pкevious part to the question I found the volume of the annular ...
2
votes
0answers
147 views

another version of log-sine integral…rather tough.

I ran across an integral with $\ln(\sin(x))$, and have not been able to make much headway. Perhaps there is no closed form, but I thought I would post it to see if anyone has some clever ideas. ...
2
votes
0answers
78 views

good books on measures and integration theory in infinite-dimensional spaces

I am looking for good books on measures and integration in infinite-dimensional spaces, covering generalizations of Wiener measure and their properties.
2
votes
0answers
82 views

Integral of a multivariate polynomial from an orthogonal basis on a constrained domain

Let $\{ \Phi_i(\mathbf{Z}) \}_{i = 0}^n$ be a basis in a functional space where $\mathbf{Z} = \{ Z_i \}_{i = 1}^m$ and $\Phi_i(\mathbf{Z})$ are multivariate orthogonal polynomials with respect to some ...
2
votes
0answers
49 views

linear DE - Where did I go wrong?

I'm trying to find the general solution for $(x+2)y' = 3-\frac{2y}{x}$ This is what I've done so far: $y'+\frac{2y}{x(x+2)}=\frac{3}{x+2}$ $(\frac{x}{x+2}y)'=\frac{3x}{(x+2)^2}$ ...
2
votes
0answers
84 views

General solution to an inhomogeneous ODE (trouble with integration by parts?)

Consider the simple inhomogeneous ODE: $$a_1 \frac{dx(t)}{dt} + a_0 x(t) = f(t)$$ By substitution, show that the following is a solution (where $\lambda_1$ solves $a_1 \lambda_1 + a_0 = 0 $, i.e. ...
2
votes
0answers
212 views

Explanation/reference for method of steepest descent (integration method, not gradient descent)

I am wondering if someone could point me to a resource where I could learn the method of steepest descent. Unfortunately, my knowledge of calculus is limited to a college multivariable calculus ...
2
votes
0answers
140 views

Convergence of Riemann-like sums

I am trying to find suitable conditions (integrability, growth...) on a function $f:\mathbb{R}\to \mathbb{R}$ such that: \begin{equation} \sum_{k\in\mathbb{Z}}f(kh)h= \mathcal{O}(1),\qquad h\to 0^+. ...
2
votes
0answers
101 views

$ \int \frac{f(x) \bar f'(x)- f'(x)\bar f(x) + g(x)\bar g'(x) - g'(x)\bar g(x) }{f^2(x) + g^2(x)} \ dx$ over $\mathbb{C}$

Evaluating $$ \int \frac{f'(x) g(x) - f(x) g'(x)}{g(x)^2} \ dx$$ should just give $\frac{f(x)}{g(x)}$. Now I have a similar quotient over $\mathbb{C}$, at least it looks similar. It's of the form ...
2
votes
0answers
253 views

Surface integral over a rectangle in Cartesian coordinates with singularity inside integration domain

I would like to integrate: $f(\rho)=\int\limits_S{\frac{1}{R}dS}$ where $R=||\rho-\rho'||_2$ and $\rho'$ represents the distance from the bottom-left corner of the rectangle. i.e. if the integral is ...
2
votes
0answers
138 views

how to calculate the integral

How to calculate the following integral: for positive constants $a_1, \cdots, a_{n+1}, $ and $i>0$ $$ \int_{S^n\bigcap\{u_m\geq 0,\ m=1,\cdots, n+1\}}\left(\sum_{m=1}^{n+1} ...
2
votes
0answers
129 views

How do I solve this integral

How do I solve this integral (expectation value) : $$\int_{-\infty}^{\infty} \psi (x)^* \hat p \psi (x)\ dx.$$ where the $\hat p =-i\hbar \frac {\partial}{\partial x}$ is an operator and $\psi (x)$ is ...
2
votes
0answers
60 views

How to compute the integral over these regions

Define $B(0,r)=\{u=(x,y,z)\in\mathbb R^3\colon|u|\le r^2\}$ for $r>0$. Let $R\subset B(0,2)$ be the region such that $|u|\le 4$ and $x\le\sqrt 3$. Let $S\subset B(0,2)$ be the region with $|u|\le ...
2
votes
0answers
131 views

Numerical Integration in Laplace domain

I need to calculate two different integrals containing a Bessel function in the Laplace domain. I have tried different kinds of methods but didn't have any luck. I don’t know how to treat the Laplace ...
2
votes
0answers
154 views

Confusion! Power series and integration

Consider the below power series: $\sum\limits_{n=1}^\infty \dfrac{x^{n}}{n^{2}}$ I know that it converges for $x\in [-1,1]$ and the sum $s(x)$ of the series is given by: $s(x) = - ...
2
votes
0answers
115 views

Finding primitives for Lebesgue integrable functions

I was wondering if there is a set of algebraic "rules" for finding primitives of Lebesgue integrals as there is one for finding primitives of Riemann integrals. I.e. for $x^{n}$ the primitive is ...
2
votes
0answers
132 views

Numerical Integration

For $r=1$, how to calculate the following integral numerically. $$\frac{8}{\sqrt{3}r^2}\int_{x=0}^{\frac{r}{2}}\int_{y=0}^{\sqrt{3}(\frac{r}{2}-x)}\prod_{i=0}^2\left(1-\frac{2}{\pi} \cos^{-1} ...
2
votes
0answers
131 views

Showing S is Jordan Measurable and Calculating the Volume

If S is the solid obtained by intersecting the ball $x^2+y^2+z^2\le4$ and $x^2+z^2\le1$ 1) How do I show that S is Jordan measurable? Can I simply say the following: "Clearly S is bounded, and the ...
2
votes
0answers
229 views

Change of variables formula

Consider an example. Let $f(x)$ be a function on the unit sphere $S^{n-1}$. Consider an integral $$ \int\limits_{S^{n-1}} f(x) \, dx $$ I want to make a substitution $x = x_{0}t + \sqrt{1-t^2}y$, ...
2
votes
0answers
707 views

Two Disk/Washer Method Problems (given a diagram)

Given a diagram from Calculus of a Single Variable by Larson and Edward (9th edition): I am interested in finding the volume of various regions when rotated about various lines. Specifically, I am ...
2
votes
0answers
54 views

Integration with multiple derivatives

Suppose I have that: $$F(x)= \frac{d H(Z)}{dZ}\cdot\frac{d^2 Z}{d x \,d y}$$ Now I want to find $$\int F(x) \, dx $$ So can I say that $$\int F(x) \, dx =\int \frac{d H(Z)}{dY} \, dZ\ ?$$ Any ...
2
votes
0answers
110 views

Integration by substitution easy question

If I have an integral $\int_{A}{u(x_1,x_2,x_3)}$ where $A \subset \mathbb{R}^3$, and I have a function $X:B \subset \mathbb{R}^2 \to A$ with $X(t_1, t_2) = (x_1, x_2, x_3)$ then I can substitute this ...
2
votes
0answers
258 views

Confused about notation and derivatives inside integrals

EDIT: To make what I am asking more clear. I've simplified it and have a more direct question. Let's say I am writing out an expression, and I want to write: $$\int_0^xF'(y)\,dy$$ However, for ...
2
votes
0answers
134 views

A proof (?) of a Cauchy Integral Theorem type claim

I want to show the following: Suppose $f\in C(|z|\leq 1)\cap O(|z|< 1)$, where $O(|z|< 1)$ means that $f$ is holomorphic in the open unit disk $D$. Then $$\int_{|z|=1} fdz=0$$ (Note: I ...
2
votes
0answers
200 views

Integral with Sums of Prime Counting Functions

I came across the following integral, while working with products of $\zeta$ primes function: $$ \int_{1}^{x}t^{-s-1} \sum_{i=1}^{\pi(t^{1/2})}\left[\pi\left(\frac{t}{p_i}\right)-i+1\right] dt, $$ ...
2
votes
0answers
105 views

Evaluating integrals over a domain in $R^n$ using space filling curves

Is there a methodology to simplify evaluation of multi-dimensional integrals using space-filling curves parameterized by a scalar parameter? I am interested in evaluating the integral of a function ...
2
votes
0answers
111 views

Approximating sums like $\sum_{j=1}^n\sum_{k=1}^{\lfloor\frac{n}{j}\rfloor}\int_1^{\frac{n}{jk}}dx$

Can anyone tell me how to approximate the following functions? $f_3(n) = \displaystyle\sum_{j=1}^n\sum_{k=1}^{\lfloor\frac{n}{j}\rfloor}\int_1^{\frac{n}{jk}}dx$ $f_4(n) = ...
2
votes
0answers
76 views

Evaluating the limit $y \to 0^+$

Given $t \in \mathbb{R}$ and $z = x + iy$ and $y>0$. $\lim_{y\to0^+} \frac{1}{t - z} = \frac{1}{t-x} + \pi i \delta(t-x)$ This limit is given in the book Integral Transforms and Their ...
2
votes
0answers
140 views

Gaussian integral where the integration limit is a linear function of a lower dimensional vector

I want to compute $$ f(x) = \int_{-\infty}^{\Lambda x} \phi_q(y; \mu, \Sigma)\ dy $$ where $p \ll q$, $x \in \mathcal{R}^p$, $\Lambda \in \mathcal{R}^{q \times p}$, $\mu \in \mathcal{R}^q$, $\Sigma ...
2
votes
0answers
68 views

Intuition about moment function derivation [OR] derivatives involving a time varying integration domain

$$ m_{{pq}}(t)=\iint\limits_{R(t)}h(x,y) dx dy $$ where $ R(t)$ the domain of integration is time varying (In fact it is the only one which is time varying). And $$ h(x,y) = x^p y^q f(x,y) dx dy ...
2
votes
0answers
244 views

How to evaluate $I_1=\int\frac{\left(\frac{a}{y}-\frac{a}{b}\right)^{1/2}}{1-\frac{a}{y}} \mathrm {d} y$?

How to evaluate this integral? $$\displaystyle I_1=\int\frac{\left(\frac{a}{y}-\frac{a}{b}\right)^{1/2}}{1-\frac{a}{y}}\mathrm {d}y$$ Here $a$ and $b$ are real constants and $y$ real variable. It is ...
2
votes
0answers
172 views

Integration by parts of $\int xf(ax)f(bx) dx$

How does one integrate $\int xf(ax)f(bx) dx$? I think it cries out for integration by parts, but I don't know how to split the integrand. Thanks.
2
votes
0answers
302 views

Seeking rationale for Hadamard's finite part of a divergent integral

I have a problem justifying the throwing away the divergent term in order to obtain Hadamard's finite part. I find this step to be highly unusual and it is not obvious to me how the resulting ...
2
votes
0answers
200 views

Geometrical interpretation of trigonometric antiderivative

I know about geometrical explanation of [defininte] integral as an area under the curve, and I wonder if there are some ideas, which may give similar insight in taking antiderivatives [indefinite ...
2
votes
0answers
202 views

When is it valid to convert a function inside a probability integral to the indicator function?

I am faced with an approximation that replaces a probability density function with the indicator function and I am at a loss as to why this is valid. We want to model the lifetime $T$ of a website ...
2
votes
0answers
321 views

Compute an integral using complex analysis techniques

Here's my problem: I have this function $\psi(\lambda)=\frac{e}{\pi\lambda}Im(e^{-\omega(\lambda-1)^{\frac{1}{4}}})$ (where $\omega=e^{-i\frac{\pi}{4}}$) defined on the interval $[1,\infty]$ and want ...
1
vote
0answers
10 views

the continuity of total variation function of a continuous function of bounded variation

Let f be a continuous function of bounded total variation (refer to http://en.wikipedia.org/wiki/Total_variation for the definition) on $[0,1]$, i.e., $\text{Var}_{[0,1]}f<\infty$. Then the total ...
1
vote
0answers
21 views

How do you calculate the Riemann zeta function of a complex number given the complex contour integral?

Can you please demonstrate how one would calculate the Riemann Zeta function of any complex number, given that the Riemann Zeta function is equal to the following (shown in ...
1
vote
0answers
35 views

Passing of the limit for Lebesgue Integral (Proof Verification)

Let $f_n\in L^1(0,1)$ and $C>0$ be such that $f_n \geq 0, f_n \rightarrow 0$ a.e., and $$\int_0^1 \max\{f_1, ..., f_n\} dx \leq C \quad \text{ for every } n.$$ Prove that $f_n \rightarrow 0$ in ...
1
vote
0answers
41 views

Integrating $\int_0^1 dx\,\ln(x-a)/(x-b)$ paying attention to cuts.

I am trying to compute the following integral, for complex $a$ and $b$ $$\int ^1 _0 dx \frac{\ln(x-a)}{x-b}$$ by turning it into something in terms of dilogarithms. But for certain values of $a$ ...
1
vote
0answers
39 views

Taking derivative under the integral sign

Reading a textbook and stuck on this one detail... would like to confirm my understanding. The book defines a function $\eta \in C^1(\mathbb{R})$ satisfying $0 \leq \eta \leq 1$, $0 \leq \eta^\prime ...
1
vote
0answers
18 views

existence and uniqueness of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
1
vote
0answers
29 views

Not lebesgue integrable function?

I want to consider the function $f:[-1,1]\times [-1,1]\rightarrow \mathbb R:f(x,y)= \begin{cases} \frac{xy}{(x^2+y^2)^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} $ And I have ...
1
vote
0answers
35 views

Evaluating an improper integral with limits $_{-\infty}^\infty$

When evaluating an improper integral with limits $_{-\infty}^\infty$, why do we need to separate the integral into $\int\limits_a^{\infty} \text{ and } \int\limits_{-\infty}^a$? My homework asked ...
1
vote
0answers
18 views

$\int_{\mathbb R^{2}} |\int_{\mathbb R} (f_{r}(t-y)- f_{r}(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0 $ as $ r\to \infty $?

Let $f\in \mathcal{S}(\mathbb R)$ with $\hat{f}$ has a compact support. For $r>0,$ put $f_{r}(x)= r^{-1}f(x/r), (x\in \mathbb R).$ We note that, $\int_{\mathbb R} |f_{r}(x)| dx = r^{-1} ...
1
vote
0answers
18 views

Limit/Integration in heat equation

While studying heat equation from PDE by L.Evans, I came across the following limit which I'm not able to prove. For $n>=1, \delta >0$ , $lim_{t \to 0+} \;\;{1 \over ...
1
vote
0answers
14 views

Change of variables formula with integrator of bounded variation

Let $G$ be be continuous with bounded variation on finite intervals. If $f$ is continuous then it is well known that $\int_a^bf(G(s))dG(s)=\int_{G(a)}^{G(b)}f(x)dx$. How general can $f$ be so that ...