Tagged Questions

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 ...

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Splitting Up Integrals and Multiplying Them

$$I_x = \int_0^b\int_0^h\rho y^2\,\mathrm{d}y\mathrm{d}x$$ So here's the current problem I'm working on, just for an example. I saw my teacher break up a triple integral in class today then multiply ...
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9 views

Volume of a solid sphere hole

A round hole of radius $\sqrt{3}$ is bored through the centre of a solid sphere of radius 2. Find the volume of the material removed . Looking for a clever way to solve this problem
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0answers
69 views

Nasty integration?

So I am trying to solve the following integral and apparently its not integrable or I might be wrong. Not even computer software can integrate. Can anyone tell me if this is integrable or not? The ...
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0answers
9 views

Integrate Two Dot-Product-Power Terms

I need to compute something like the following integral:$$ \int_{\Omega_\vec{a}} \left< \vec{x} \cdot \vec{b} \right>^n \left(\vec{x}\cdot\vec{a}\right)^m d\vec{x} $$Notational issues: The ...
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0answers
18 views

Radon transforms and determining a separable function

I am interested in the Radon transformation for separable functions $F(x,y) = f(x)g(y)$. Why is it in tomography that a separable function is determined completely by two of its projections ? And ...
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0answers
8 views

Integrate determinant of product of two matrices

Let $V\left(i,j\right) = \alpha_j^{i-1}$ be the $\left(i,j\right)^{th}$ element of the matrix $V\in\mathbb{R}^{n\times n}$. Such matrices are called Vandermonde matrices. Let $X = \left|V\times ...
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2answers
44 views

Why is $F(x)$ continuous at $x=0$?

Let $$f(x) = \left\{ \begin{array}{ll} x & \mbox{if } x < 0 \\ \sin x & \mbox{if } x \ge 0 \end{array} \right.$$ $F(x)$ the anti-derivative should be $\frac{x^2}{2} + C_1$ for ...
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4answers
43 views

Derivative of $\int_{x}^{0} \frac{\cos xt}{t} dt$

I am working on the following problem: Find the derivative of $f(x)=\displaystyle \int_{x}^{0}\displaystyle\frac{\cos xt}{t}dt$. The answer I am supposed to get is $\displaystyle ...
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1answer
15 views

Volume of region inside a surface

Find the volume of the region inside the surface $z = x^2 + y^2$ and between $z = 0$ and $z = 10$. Really the only thing I need help with in this problem is setting up the limits of integration. ...
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0answers
20 views

Finding an integral for a given Riemann Sum

Take the Riemann sum: $= \displaystyle \lim_{m\to\infty} \frac{1}{m}\sum_{x=1}^{m} me^{-x}$ How can someone convert that into an integral? We know $\Delta(x) = \frac{1}{m}$. So $me^{-x}$, is the ...
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1answer
17 views

Nonconvergent convolution integral

I have a convolution integral where $F(\tau) = F_0$ and $g(t - \tau) = \sin(\omega_n(t-\tau))$ so $$ F_0\int_{t_0}^{\infty}\sin(\omega_n(t-\tau))d\tau $$ which doesn't converge. Can I do the ...
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0answers
13 views

Double Integral Mistake with Parametric Equation

I'm trying to figure out the mass of an object bounded by $y=0$ and $y=\sqrt{1-x^2}$ the density at a given point is proportional to its distance from the origin; $\rho(x,y) = kxy$. So I set it up ...
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0answers
60 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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1answer
23 views

Numerically solve integral with a function as variable of integration

I want to use a function as variable of integration for example in evaluating the integral: $\int_0^1 e^{\cos x}f(\sin x)d\cos x$ in which $f(x)$ is an arbitrary function.
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0answers
16 views

Integeration of 2nd derivative

I have a question regarding solving the integral of the partial. Here is the equation: $$R_m = \int \psi^m \frac{\partial^2}{\partial\psi^2} \left[\left\langle \epsilon_\phi | \psi \right\rangle ...
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1answer
40 views

Why am I obtaining an imaginary part for my integration

I try to solve an integration as follows, $$\int \frac{sy^{-1}}{(1+sy^{-1})} \text{exp}(-\sqrt{y})dy$$ as you can see its pretty complicated, and I get an answer like the following using Wolfram ...
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1answer
29 views

Spectral Measures: Domain Criterion

Given a topological space $\Omega$ and a Hilbert space $\mathcal{H}$. Let $\mathcal{B}(\Omega)$ be its Borel algebra and $\mathcal{B}(\mathcal{H})$ its bounded operators. Moreover, given a spectral ...
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2answers
19 views

Why is $\vec{s}=\frac{\vec{r}}{V^\frac{1}{3}} \Leftrightarrow d\vec{s}=\frac{d\vec{r}}{V}$?

I am following a course which contains a part in statistical thermodynamics. One of the questions involves the partition function $Q_N$. I could not figure out the answer of the question myself, so I ...
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1answer
97 views

Evaluating by real methods $\int_0^{\pi/2} \frac{x^5}{2-\cos^2(x)}\ dx$

I'm sure you guys can briefly get the result by some methods of complex analysis, but now I'm only interested in real analysis methods of proving the result. What would you propose for that? ...
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0answers
17 views

Asymptotic expansion at infinity of integral function

Given $q\in(0,1)$ find $z$ such that $$ F(z)\equiv\int_{-\infty}^{z}\frac{e^{-\frac{y^2}{2 \sigma _{22}^2}} \text{erfc}\left(\frac{\rho \sigma _{11} y-\sigma _{22} V}{\sqrt{2-2 \rho ^2} \sigma ...
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2answers
35 views

meaning of integration

I read that integration is the opposite of differentiation AND at the same time is a summation process to find the area under a curve. But I can't understand how these things combine together and ...
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0answers
16 views

surface and cone integrals [on hold]

can someone take me through these two questions, I have the answers but not the steps and I have no idea how to even get started, thanks!
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1answer
37 views

Integrating $\int \sec^2(x) \tan(x) dx$ by trig substitution

I know I am supposed to integrate $$\int \sec^2(x) \tan(x) dx$$ by substituting $u = \tan(x)$ and get $du = \sec^2(x)$. However, why can't I use $u = \sec(x)$, $du = \tan(x) \sec(x)$?
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1answer
12 views

Poisson integral with discontinuous $U$

Let $U$ be a piecewise continuous function and bounded for all real numbers. Then define the Poisson Integral for the UHP to be (It can be deduce from the one for the unit circle). ...
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2answers
22 views

Proving complex integral on jordan region boundary equals to zero

Let $D\subset\mathbb{C}$ be a region bounded by jordan curve $\gamma$. Prove that: a. $\int_\gamma z \, dz=0$ b. $\int_\gamma \bar{z} \, dz\neq0$ (hint:$\bar{z}\,dz=(x-iy)(dx+i\,dy)$) ...
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2answers
30 views

U-Substitution. Why do you multiply the integrand by -1 in this case?

$$\int_0^{\pi/2} \! \frac{\sin x\cos x}{(4-\sin^2 x)^2} \:\text{d}x$$ set $u = 4-\sin^2 x$, therefore $du = -2 \sin x \cos x \text{d}x $ $$-\frac{1}{2} \int u^{-1/2} \text{d}u $$ Change the range ...
1
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1answer
51 views

Tough definite integration

For a curve given by: $x=e^{-t}\cos{2t}$, $y=\sin t$ R is the region bounding this curve, the x axis and the y axis (y-intercept is point a and x-intercept is point b). Find the exact coordinates ...
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2answers
31 views

How integrate $ \iint_{D} (\frac{x^2}{x^2+y^2})dA, \ \ \ \ D: x^2+y^2=a^2 \ \ and \ \ x^2+y^2=b^2, \ \ 0<a<b $

I'm trying to resolve this integral $$ \iint_{D} (\frac{x^2}{x^2+y^2})dA, \ \ \ \ D: x^2+y^2=a^2 \ \ and \ \ x^2+y^2=b^2, \ \ 0<a<b $$ I tried with polar coordinates: $$ x = r\cos{\theta} \\ ...
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2answers
16 views

solve polar coordinate integral

Evaluate $$\int_0^R\int_0^\sqrt{R^2-x^2} e^{-(x^2+y^2)} \,dy\,dx$$ using polar coordinates. My answer is $-\frac{1}{2}R(e^{-R^2+x^2}-1)$ but I want to confirm if that's correct And also, when I ...
3
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0answers
56 views

Solving double integrals numerically?

I have written this in way to make it as much as possible non-confusing. I will start describing my problem and I will walk you through my question, I have a double integration which I am trying to ...
3
votes
3answers
36 views

Evaluating indefinite integral using a trigonometric substitution

I have this integral: $$\int\frac{x^3}{\left(\sqrt{4x^2+9}\right)^3}\,dx$$ I tried to solve it with a trigonometric substitituon but I can't get any result. I would appreciate if somebody could help ...
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1answer
17 views

Arc length of this function

It is given that $x^2=(2y)^2$, he asks to give the arc length of this function, $1\leq x \leq 2\sqrt2$. Answer is $1/27 (19^{3/2} - 10^{3/2})$.
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0answers
14 views

Norm of arbitrary constant

I'm sitting in front of an exercise (basics in quantum mechanics), which wants me to check if the integral of a given function can be normed. One of those functions is the integral of zero from ...
4
votes
1answer
48 views

Calculating indefinite integral?

I want to calculate $$I_n = \int \frac{d\theta}{\sin^n(c\theta)\cdot \cos(c\theta)}. $$ The answer is $$-\frac{1}{c(n-1)\sin^{n-1}(c\theta)}+ ...
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0answers
23 views

Surface integral defined by a closed curve

So I know how to integrate over a surface defined by a parametric equation $$ \textbf{r}(u, v) = x(u, v) \textbf{i} + y(u, v)\textbf{j} $$ But what if the surface is defined as the area inside a ...
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0answers
29 views

integral sulotion over a and t [on hold]

what is the solution of this integral:$$\int^1_0 \frac{-2(t+a)+(1-a)}{((t+a)^2+(1-a)^2)^2} dt$$ canyou help me? that is a part solultion of a question which I should to solve it!
4
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0answers
86 views

How to compute or simplify this nasty integration?

Any hints on solving an integration of the following form, $$\int_{x}^{+\infty}\left(1-\frac{1}{1+sy^{-1}}\right) \left(\text{exp}(-\sqrt{y})+ y^{-\frac{1}{2}}(1-\text{exp}(-\sqrt[4]y)\right)dy $$ ...
2
votes
1answer
29 views

Gauss-Green Theorem from generalized Stoke's Theorem.

I am trying to deduce the next identity (Green-Gauss theorem) $$\int_\Omega \dfrac{\partial u}{\partial x_i} dx = \int_{\partial \Omega} uv_i dS$$ from the generalized Stoke's theorem for manifolds. ...
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0answers
8 views

lim Ln delta and d what do these mean? [on hold]

I need to know what does lim and Ln mean? And the difference between delta x and dX .I need these stuff in physical chemistry.
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2answers
29 views

Is the following true regarding integration?

Is the following correct? $$ \int_{x}^{+\infty} \left(f(u)-g(u)\right) du = \int_{x}^{+\infty} f(u)du - \int_{x}^{+\infty} g(u) du$$
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1answer
40 views

Prove the volume of a ball with radius approaching 0

Let f be continuous and let Br be the ball of radius r > 0 centered at $(x_0, y_0, z_0)$. Let V (Br) be the volume of Br. Prove that $$\lim_ {r\to0} \frac{1}{V(Br)}\ \iiint_{Br} \ f(x,y,z) dV = f(x_0, ...
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1answer
25 views

asymptotic expansion of this integral

How to get the asymptotic expansion for this integral $\int_{0}^{1}\exp(-x/t)dt $ in the limit $x\rightarrow 0$ ? I took $x/t=u$ and did integration by parts (IP) but if I keep doing IP, I get a ...
1
vote
1answer
33 views

Changing to spherical coordinates to evaluate the integral

$$\iiint_D \,dz\,dy\,dx$$ where the region $D$ is defined as followed: $$0<z<\sqrt{9-x^2-y^2}$$ $$0<y<\sqrt{9-x^2}$$ $$0<x<3$$ I got the corresponding spherical coordinates for ...
4
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0answers
47 views

Sorting out some integrals from physics

I'm doing some physics for a change, and I'm trying to sort things out a bit. From the definitions of mass, torque, momentum and angular momentum I've come up with the following integrals: ...
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0answers
17 views

Replacement in integral

Let's have integral $$ \int \limits_{-\infty}^{\infty}d^{4}k f\left(k^{2}, (k \cdot p )\right)k_{\mu} k_{\nu}, \quad d^{4}k = dk_{1}dk_{2}dk_{3}dk_{4}. $$ Here $f$ is integrable function, $(k \cdot p ...
0
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1answer
32 views

Proper definite of riemann integral (limit version)

I am sort of confused. Suppose we are given the series, $\displaystyle \lim_{n\to\infty}\sum_{k=1}^{n} \frac{k^{99}}{n^{100}}$ How can this be written as an integral, and what would the variable ...
3
votes
2answers
69 views

Solve $ \int_{0}^{\infty} \sin^2 \left(\frac{1}{x}\right)\mathrm{d}x$

I think this integral does not converge. I want to estimate downward the integral, but don't know how to.
5
votes
1answer
51 views

using complex or real analysis solve $\int_{0}^{\pi/2}\frac{x^m}{\sin x}dx$

closed form for $$\int_{0}^{\frac{\pi}{2}}\frac{x^m}{\sin x}\ dx$$ I slove it for some m but in general i failed. I tried by part , by substitution,by using $\sin x =\frac{e^{ix}-e^{-ix}}{2i}$ . I ...
2
votes
1answer
23 views

Integration of $x^a$ and Summation of first $n$ $a$th powers

I'm learning some discrete mathematics. I already knew a little (very little) calculus, and I noticed something. I think it's just a coincidence, so I'm sorry if this is a bad question. There are some ...
2
votes
2answers
53 views

Finding the $nth$ partial sum for $e^{-n}$

Here is the question: $$\displaystyle \sum_{n=1}^{\infty} e^{-n}$$ Instead of using the formula of $\large\frac{1}{1-r}$ I want to try to get the partial sums. $S_1 = e^{-1}$ $S_2 = e^{-1} + ...