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)

1
vote
2answers
82 views

Complex value for volume, using triple integrals

I'm trying to calculate the volume a hyperboloid, within $$z=0$$ and $$z+\frac 12 x-3=0.$$ The hyperboloid: $$x^2+\left(\frac y2\right)^2-z^2=5.$$ I calculated the projections on $xz$, $yz$, to use ...
1
vote
1answer
28 views

Calculation of triple integral over an elliptic cone

The task at hand is to calculate the following integral: $$I=\iiint_V z^2\,dx\,dy\,dz$$where$$V=\{(x,y,z)\in\mathbb R^3:\,\frac{x^2}{a^2}+\frac{y^2}{b^2}\le\frac{z^2}{c^2},\, 0\le z\le h\}$$ Using the ...
1
vote
1answer
37 views

Interpretation of Line Integral with respect to discrete variable

In the paper I am reading, (http://arxiv.org/abs/1308.5376), they solve an integral and I am trying to replicate the results. This question is a simplified version of the integral they calculate, I ...
6
votes
2answers
162 views

Determining $\int \frac{(x-1)\sqrt{x^4+2x^3-x^2+2x+1}}{x^2(x+1)}dx$

$$\int \frac{(x-1)\sqrt{x^4+2x^3-x^2+2x+1}}{x^2(x+1)}dx$$ Attempt: Simplification of the root factor: ...
10
votes
1answer
268 views

Integral formula for $\int_{0}^{\infty}e^{-3\pi x^{2}}((\sinh \pi x)/(\sinh 3\pi x))\,dx$ by Ramanujan

Towards the end of G. N. Watson's (one of the joint authors of famous book "A Course of Modern Analysis") paper "The Final Problem: An Account of the Mock Theta Functions" the following formula of ...
0
votes
3answers
49 views

Uniform convergence and integration: $\lim_{n \to \infty} \int_{0}^{1} \frac{1}{1+x^2+\frac{x^4}{n}} dx=?$

For f$_n$(x)=$\frac{1}{1+x^2+\frac{x^4}{n}}$, we need to calculate $\lim_{n \to \infty} \int_{0}^{1} f_n(x) dx$ . I want to prove f$_n$ is Riemann integrable and f$_n$ uniformly converges to f, then ...
0
votes
1answer
52 views

Find the centroid bounded by $x+y = 2, y=x^2, y=0$.

I found the intersection point $(-2,4)$ and $(1,1)$ I decided to make everything in terms of $y$. Thus, $y=x^2$, $y = 2-x$. So I will use the integral bound of -2 to 1. $2-x$ is the higher curve, ...
1
vote
0answers
28 views

Let $f:[-2,5] \to \mathbb{R}$ integrable such as $f(x) \neq 0$ for all $x \in [-2,5]$, therefore $\frac{1}{f}$ is integrable in $[-2,5]$. T/F?

Let $f:[-2,5] \to \mathbb{R}$ integrable such as $f(x) \neq 0$ for all $x \in [-2,5]$, therefore $\frac{1}{f}$ is integrable in $[-2,5]$. I think that is true. Let's go for it. Let $g:I \to ...
2
votes
2answers
70 views

Integrating a First Order Differential Equation (The West Equation)

I am currently doing a project about Growth and have found this really interesting Math Model by Dr. Geoffrey West et al in 2001 while researching. The paper can be found at this link. I was ...
1
vote
0answers
47 views

Numerical integration on log binned samples

I would like to do numerical integration on the function below to get $\omega(r_{p})$ (original image) $$ \omega(r_p) = 2\int_{0}^{\infty} \xi(r_p,\pi)\;\mathrm{d}\pi $$ I have values for ...
3
votes
5answers
220 views

Why is it not true that $\int_{-\infty}^\infty{\rm} x\,dx=0 \, $ given that x is an odd function?

I was given this to do at my own will and look at why it is divergent, but to be sincere I have no idea on how I can start approaching it. I just need explanation to why the answer is 0. I'm studying ...
1
vote
2answers
79 views

How do you take the integral of $\int \frac{\mathrm{d}y}{y(3-y)}$?

How do you evaluate the following? $$\int \frac{\mathrm{d}y}{y(3-y)}$$ I have looked at the solution and I don't understand how they are taking the integral of this? They go from: $$\int ...
2
votes
3answers
170 views

What is the integration of $\int 1/(x^{2n} +1)dx$?

I am a student who is preparing for IIT exam. I was just practicing calculus and encountered this problem. I tried different substitutions but none of them seemed to work. So what is the integration ...
0
votes
1answer
39 views

integral by parts and relation with $I_1$

Compute $I_n=\int \frac 1 {(x^2-1) ^n} \Bbb d x$. My work: $I_1 = \int \frac 1 {x^2-1} \Bbb d x= \frac 1 2 \int (\frac 1 {x-1} -\frac 1 {x+1}) \Bbb d x = \frac 1 2 \ln (x-1) - \frac 1 2 \ln (x+1)$
3
votes
6answers
74 views

$\int_a^b f = \int_a^b g$, then there exists a point $c \in [a,b]$ such that $f(c) =g(c)$.

Let $f : [a,b] \to \mathbb R$ and $g : [a,b] \to \mathbb R$ be continuous on $[a,b]$ and $\int_a^b f = \int_a^b g$, then there exists a point $c \in [a,b]$ such that $f(c) =g(c)$. If we let $h(x) = ...
1
vote
0answers
40 views

Line Integrals, dx, dy and parameters

So I posted a similar question the other day, which I answered myself since I was under the impression that I figured this stuff out. But today I found a question where my technique broke down and I ...
2
votes
1answer
52 views

Determining position with respect to time

If we assume $s(t)$ as a time-dependent position function and $v(t)$ as a time-dependent velocity function($v = \frac {\mathrm{d}s} {\mathrm{d}t}$) and $v=8\sqrt{s}$, how could I determine $s$ with ...
2
votes
2answers
83 views

If $\{f(x)\}^2 = 2\int_0^xf(t)dt $ then $f(x) = x$ for all $x \geq 0$.

A function $f$ is continuous for all $x \geq 0$ and $f(x) \neq 0$ for all $x >0$. If $\{f(x)\}^2 = 2\int_0^xf(t)dt $ then $f(x) = x$ for all $x \geq 0$. But I am stuck with the sum.
0
votes
0answers
34 views

Integral on an region defined by a regular grid of points

I'm trying to evaluate a multidimensional integrand $f$ that I know the major contribution is restricted to a specific region around its maxima (to be concrete, imagine a 2D gaussian function). What ...
5
votes
1answer
71 views

A product version of Riemann integral

Motivated by Riemann sum in Riemann integral and motivated by relations between infinite series and infinite products we ask: Assume that $f:[0, 1]\to \mathbb{R}$ is a positive function. Assume ...
0
votes
1answer
239 views

Surface area of $y = \sin(\pi x)$, from $x=0$ to $2$, rotated about the $x$-axis.

When I use the surface area formula I get 0, and Wolfram got zero as well when I use the bounds 0 to 2, why is this? However the solution manual uses the integral with bounds 0 to 1.. What is going ...
-3
votes
2answers
44 views

To show that the function $f(x) = x[x], x \in [0,3]$ is integrable.

To show that the function $f(x) = x[x], x \in [0,3]$ is integrable and the evaluation of the integral $\int_0^3f$ cannot be done by the fundamental thorem of Integral Calculus. I am having confusion ...
1
vote
1answer
49 views

Derivative of integral and variable substitution

My question is about the validity of this identity and if there is some error in my argument: $$\int_0^{\infty}\frac{d}{dt}f(t-x)dx = -\int_0^{\infty}\frac{d}{dx}f(t-x)dx$$ The argument goes as ...
2
votes
1answer
103 views

Integral of $\sqrt{4-\sin^2(x)}$

So we have to find the antiderivative of $\sqrt{4-\sin^2x}$. What I did was to put $\sin x=t;\cos xdx=dt$. But now I am not able to calculate the integral of $\sqrt\frac{4-x^2}{1-x^2}$. So how to ...
1
vote
1answer
88 views

Quadrature formula on triangle

I am looking for a quadrature formula on the triangle, with points at the vertices and at the mid-edges, so 6 points, and that is exact for polynomials of degree at least 2, with weights strictly ...
10
votes
3answers
363 views

Evaluating the limit of a certain definite integral

Let $\displaystyle f(x)= \lim_{\epsilon \to 0} \frac{1}{\sqrt{\epsilon}}\int_0^x ze^{-(\epsilon)^{-1}\tan^2z}dz$ for $x\in[0,\infty)$. Evaluate $f(x)$ in closed form for all $x\in[0,\infty)$ ...
2
votes
1answer
45 views

Does the formula for arc length hold for other coordinate systems?

Does the formula for arc length, integration of $\sqrt{(x'(t))^2 + (y'(t))^2 + (z'(t))^2}$, hold for other coordinate systems, such as cylindrical coordinates, meaning can I compute the integral of ...
3
votes
1answer
187 views

Singularities of an integral

We have the integral : $$I(t)=-i\int_0^\infty \frac{\log\left[\frac{\sin(t\log\sqrt{1+ix})}{\log(1+ix)} \right ]-\log\left[\frac{\sin(t\log\sqrt{1-ix})}{\log(1-ix)} \right ]}{e^{2\pi x}-1} \, dx$$ I ...
0
votes
1answer
84 views

How to simplify this integrand,

I am trying to compute arc length in three dimensions but am currently stuck with integrating $$\sqrt{1+ e^{-2t} + 4e^{-2t}}$$ Can I get some hints on how to simplify? I didn't combine the second ...
0
votes
1answer
76 views

Gauss divergence theorem, $div(F) = 0$?

I'm trying to solve the following problem using the Gauss divergence theorem. I have to calculate the Flux of $$ f(x,y,z) = (\sin(yz),y+\sqrt{x^2 + z^2}, 1-z) $$ through the surface $$ \Omega = ...
0
votes
3answers
71 views

Another way to compute the length of a curve

What is the length of C, where C is the graph of the function $$f(t) = \frac{e^t + e^{-t}}{2}$$ on the interval $[0,2\pi]$. Is there a nice way to compute this arc length integral, without knowing ...
2
votes
1answer
65 views

Integration on 4th dimension unit ball

I'm trying to calculate \begin{equation*}\int _B e^{x^2+y^2-z^2-w^2}\end{equation*}where \begin{equation*}B=\{\vec x\in\mathbb R^4:||x||\le 1\}\end{equation*}is the unit ball in 4 dimensions. I tried ...
2
votes
2answers
70 views

I am having trouble solving this problem from the book “ Measure Theory” by Donald L.Cohn.

Let $(X,\mathcal A ,\mu)$ be a measure space and let $f$ and $f_1 ,f_2 ,....$ be non-negative functions that belong to $\mathcal L^1(X,\mathcal A,\mu,\mathbb R)$ and satisfy- (i) $\{f_n\}$ converges ...
1
vote
2answers
85 views

Evaluate the Integral : $\int_{2}^{1}\frac{dt}{8-3t}$

$$\int^2_1\frac{\mathrm{d}t}{8-3t}$$ The Fundamental Theorem of Calculus: Suppose $f$ is continuous on $[a,b]$. If $g(x) =\int^x_0 f(t)\ dt$, then $g'(x)=f(x)$ $\int^b_a\ f(x)\ ...
1
vote
0answers
44 views

Book to learn Darboux integral

What are some good references to , good book to learn Darboux integral ( https://en.wikipedia.org/wiki/Darboux_integral ) ? Please help .
2
votes
0answers
40 views

Fourier Cosine Expansion of Piecewise Continuous Function

Hi I am trying to represent this following function: $$f(x)=\begin{cases} 35.6236 + 0.161087e^{59.9842x},0\leq x < 0.1 \\ 35.6236 + 0.161087e^{59.9842 (-x + 0.2)},0.1\leq x \leq 0.2 \\ ...
0
votes
2answers
83 views

Solving this inequality with integral

We have function $f:\mathbb{R}-\{2 \}\to\mathbb{R}$ $$f(x)=\frac{x^2}{x-2}$$ Show that $8\le\int\limits _3^4f\left(x\right)dx\le9$ I solved the definite integral and got $\int\limits ...
0
votes
0answers
16 views

An integral from the integral geometry about the isoperimetric inequality.

The problem is from the book "Integral Geometry and Geometric Probability" by Santalo (1976), Chapter 1.3.5, Notes and Exercises (page 37). Given a convex closed curve $C$. Let $A_1$, $A_2$ be the ...
2
votes
1answer
47 views

Example comparing Riemann's and Lebesgue's methods of integration

It is well known that a function which is Riemann integrable is also Lebesgue integrable, and both integrations result in the same value. Question: Can one give an example of a Riemann integrable ...
0
votes
1answer
48 views

Matrix Multiplication, Trace and Integration

Let $\omega(x)$ be a $p\times 1$ vector-valued function defined on a random variable $X$ with CDF $F$. Now define $$V:=\int \omega(x)[\omega(x)]^T dF(x).$$ Then define $\gamma$ as follows. $$ \gamma ...
0
votes
0answers
17 views

Primitive-ability of a function

Prove that the function $f:{R}\to{R}$ is primitive-able(does this term exist in English?) and find one of its primitives. $$f(x)=\left\{\begin{array}{cc} 1-x & x<1 \\ x^2-2x+1 & x \geq 1 ...
0
votes
1answer
79 views

Problem about integration

Let $\mathcal R$ be a $\sigma$-algebra in a nonempty set $X$, let $\mu$ be a positive measure on $\mathcal R$, let $f:X\to \mathbb C$ be measurable relative to $\mathcal R$,and $f\in L^1(\mu)$. Let ...
0
votes
1answer
82 views

Closed form for an integral

I am trying to find a closed form for this integral: $\int\limits_{a}^{\infty} \exp(-\frac{b}{x})\exp(-cx)dx$ where a,b,c, are positive constants. Does anyone have any suggestions or can advise? ...
0
votes
0answers
29 views

Convergence of Laplace transform and its inverse

There is a sequence of functions $F^{\epsilon}(\lambda)$ which converges to 0 as $\epsilon \rightarrow 0$. Assume that each $F^{\epsilon}(\lambda)$ has a inverse Laplace transform f(s) such that ...
6
votes
0answers
81 views

Fredholm integral?

If one exists, find a continuous, bounded function $f: \mathbb{R} \to \mathbb{R}$ which is not identically zero and which satisfies$$0 = \int_0^\infty K(t, s)f(s)\,ds$$for all $t \in \mathbb{R}$, ...
1
vote
2answers
59 views

Show that $\int_0^\infty e^{-x}{\sqrt x}dx=\frac{\sqrt\pi}{2}$ by using $\int_0^\infty e^{-x^2}dx=\frac{\sqrt\pi}{2}$

I'm trying to do integration by parts to be able to use $\int_0^\infty e^{-x^2}dx=\frac{\sqrt\pi}{2}$, but is not working.
4
votes
2answers
68 views

What happens when I convert a Taylor series into an integral?

Suppose we have the Taylor series of an analytic function: $$f(x) = \sum_{k=0}^\infty \frac{1}{k!} a_k x^k$$ Then I decide to (kind of) turn it into an integral: $$g(x) = \int_0^\infty ...
1
vote
2answers
65 views

Prove convergence of $\int_1^\infty \frac 1 {x(\sqrt x + 1)} dx$

Prove the convergence of $\int_1^\infty \frac 1 {x(\sqrt x + 1)} dx$ This was a question on an exam. I needed to prove that the above integral converges using the comparison test. I thought about ...
3
votes
2answers
91 views

Does anyone know of a closed form solution to the following integral?

Does anyone know of a closed form solution to the following integral? $$ \DeclareMathOperator\erf{erf} \newcommand{d}{\;\mathrm{d}} \int^{+\infty}_{-\infty} \erf^{\;m}\!(x) \frac{\d^n ...
1
vote
1answer
31 views

Question about the limits of definite integrals

Let me take an example that I've come across while studying Fourier series, We all know that $$\int_{-a}^{a} \sin \left( \frac{n\pi x}{a} \right) dx = 2 \int_{0}^{a} \sin \left(\frac{n \pi x}{a} ...