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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0
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1answer
20 views

Am I correct with this change of variable?

I have been solving a problem from a paper I read related to poisson point processes and for some reason I am not reaching the same result the paper has. The problem is re-expressing an expression by ...
0
votes
1answer
19 views

Find the volume of the solid generated by revolving the region bounded by $y=x$ and $y=x^2$ about the line $y=x$

Find the volume of the solid generated by revolving the region bounded by $y=x$ and $y=x^2$ about the line $y=x$ I am confused, how do we approach such problems, where the rotation lines are not ...
0
votes
1answer
20 views

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 ...
0
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1answer
19 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
0
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0answers
81 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 ...
1
vote
0answers
11 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 ...
0
votes
1answer
22 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 ...
0
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0answers
10 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 ...
0
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0answers
4 views

Saddle point method for asymptotic expansion

Any idea on how to derive an asymptotic expansion of the following integral for $x$ around zero (using Saddle point method): $$\int_{0}^{x}\left(u+\alpha\right)^{-a - \left(b-a\right)e^{-\lambda ...
1
vote
2answers
48 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 ...
1
vote
4answers
49 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 ...
0
votes
1answer
17 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. ...
0
votes
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 ...
0
votes
1answer
18 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 ...
0
votes
0answers
15 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 ...
2
votes
0answers
62 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 ...
0
votes
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.
0
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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 ...
2
votes
1answer
45 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 ...
1
vote
1answer
33 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 ...
2
votes
2answers
20 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 ...
10
votes
1answer
104 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? ...
0
votes
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 ...
1
vote
2answers
36 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 ...
-1
votes
0answers
19 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!
3
votes
1answer
39 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 $u = \sec(x)$, $du = \tan(x) \sec(x)$?
0
votes
1answer
16 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). ...
0
votes
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:$\bar{z}\,dz=(x-iy)(dx+i\,dy)$) ...
0
votes
2answers
35 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
vote
1answer
52 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 ...
0
votes
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} \\ ...
1
vote
2answers
17 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
votes
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
39 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 ...
0
votes
1answer
18 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 $1/27 (19^{3/2} - 10^{3/2})$.
0
votes
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)}+ ...
0
votes
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 ...
0
votes
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
votes
0answers
87 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
33 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. ...
0
votes
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.
0
votes
2answers
29 views

Is the following true regarding integration?

Is the following correct? $$ \int_{x}^{+\infty} \left(f(u)-g(u)\right) du = \int_{x}^{+\infty} f(u)du - \int_{x}^{+\infty} g(u) du$$
0
votes
1answer
46 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, ...
0
votes
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
votes
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: ...
0
votes
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
votes
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.