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 ...

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104 views

Changing the order of integration in the proof that Laplace maps convolution to multiplication

I was reading the proof that Laplace transform maps the convolution of two functions to the multiplication of their transforms. Or mathematically $$\mathcal{L}[f*g]=\mathcal{L}[f]\,\mathcal{L}[g],$$ ...
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53 views

Interpretation of a line integral with respect to x or y .

i read about some interpretation ideas in Interpreting Line Integrals with Respect to $x$ or $y$ and i was wondering if the interpretation given below is right or not ? informally put can we say ...
2
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80 views

A Book recommendation for double Integrals?

I have a really hard time learning Double Integrals, which I attempted to understand when I first saw the use of polar co-ordinates for Integrals. So my goal is to learn double Integrals and also ...
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136 views

Choosing a contour to integrate over.

What are the guidelines for choosing a contour? For example to integrate a real function with a singularity somewhere. What type of contour from Square, keyhole, circle, etc should be chosen for ...
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15 views

Bounding $\int_{\infty}^{\infty}|g(s)v^3k(v)|dv$ where $k$ is a second-order kernel

Suppose $k$ is a nonnegative, bounded real-valued function that satisfies $$ \int_{-\infty}^\infty k(v)dv=1,\quad k(v)=k(-v),\quad \int_{-\infty}^\infty ...
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29 views

Solving String Vibration Using Integral Transform

$$U_{tt} - c^2 U_{xx}= -g$$ where BC: $U_{x}(0,t)=a\sin(ωt)$ IC: $U(x,0)=0$, $U_{t}(x,0)=0$ where $c, g, A$ and $ω$ are positive constants Normally I wouldn't post for help here but I am ...
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36 views

$E(g(X)), E(g'(X)) <\infty $ implies $\lim_{x\rightarrow \infty} f(x)g(x)= 0$ ($f$ is the density of $X$)?

I am trying to figure out the Stein's identity which asserts that for r.v $X$ having pdf $$p_\theta(x)=\exp\{ \theta T(x)-A(\theta)\}h(x)$$ where $ T$ is differentiable and $g>0$ is ...
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81 views

Using residue theorem along a branch cut to evaluate the inverse Laplace transform

I am trying to find the inverse Laplace transform of $f(z)$ using the residue theorem. Can you please check to see if what am doing below is correct? I am not really sure about what I am doing. ...
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39 views

Determining solid region from bounds of triple integral

If you have an integral such as: $$\int_0^1\int_0^{2-x^2}\int_0^{2-x}f(x,y,z)dydzdx$$ How can you determine the equation for the solid region represented by the bounds of this triple integral? Does ...
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97 views

How to show that a piecewise constant function is integrable, using the upper and lower sums?

Let $f(x) = \begin{cases} 1 &\mbox{if } 0\leq x<1 \\ 3 &\mbox{if } 1\leq x<2 \\ 2 &\mbox{if } 2\leq x\leq 3. \end{cases}$ Show that $f(x)$ is integrable by $(a)$ ...
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45 views

Evaluating the integral $\int_0^a x^{v/2} e^{-\alpha x} J_v(2\beta\sqrt{x}) dx$

I'm searching for a way to evaluate the following integral: $$\int_0^a x^{v/2} e^{-\alpha x} J_v(2\beta\sqrt{x}) dx$$ where $J_v(x)$ are the Bessel-functions, and $v \in \mathbb{N}, (a,\beta) \in ...
2
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59 views

Value of improper Integral

I need help in finding the value of the integral $$\displaystyle \int_0^\infty \left(\frac{x^2}{1+x}\right)^{n-1}e^{-tx}dx,$$ where $n$ is a positive integer and $t$ is a positive real number.
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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 ...
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91 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 ...
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38 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 ...
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46 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 ...
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35 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 ...
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27 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 ...
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33 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 ...
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60 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 ...
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41 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 ...
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128 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 ...
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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 ...
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157 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 ...
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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 $\mu_y=0$ ...
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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 ...
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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 $a\in\mathbb{R}$ $$ ...
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95 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 $\int_0^1\frac{\ln(x)\ln(1+x^2)}{1+x}dx$ ...
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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 ...
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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
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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 ...
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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 \\ &= ...
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35 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) \, ...
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201 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 ...
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25 views

$f(x,y)={1 \over x^2} \sum_{n=1}^{\infty}{\int_x^y{\sqrt{t} \over {1+ ({t \over x} -n)^2}}} dt$ is differentiable?

Let $ D=\{(x,y) \in \mathbb{R}^2 : x>0, y>0\}$. Show that the function $$f(x,y)={1 \over x^2} \sum_{n=1}^{\infty}{\int_x^y{\sqrt{t} \over {1+ ({t \over x} -n)^2}}} dt$$ is well defined on $D$. ...
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71 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 ...
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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 ...
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35 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
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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: $$ (\sum_{r=1}^{\infty}\frac{z(r)}{r})\times \int_0^\infty f(x) dx = \lim_{h \rightarrow 0} \sum_{i=1}^{n} f(k_ih)h$$ ...
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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 ...
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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 ...
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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 ...
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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
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52 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
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85 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
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68 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
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326 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
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175 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
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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
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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 ...