Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [integration]

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 the integrals being considered. This tag often goes along with the (calculus) tag.

0
votes
1answer
16 views

Find the area of surface of revolution

$y^2 = 1+x, 0 \leq x \leq 3$, rotated about the x-axis. This is really frustrating because the answer is wrong, and after redoing this problem, I get an answer that is still wrong, but different from ...
-2
votes
0answers
22 views

Find the volume of a solid

Consider the solid bounded by 4$x^2$+$y^2$+$z^2$ = 9 and z$\ge$ $\sqrt{4x^2+y^2}$ Hint: Use and apply symmetry. Find the volume of the solid. I tried to convert the equations to spherical ...
0
votes
1answer
29 views

How to solve this integration or does this have a closed form?

The integral I am dealing with is below. I need to find the closed-form expression of this integral. $$\int_0^\infty \ln\left(1+\frac{A}{1+B+Cx}\right)\frac{e^{-x/M}}{M}\,dx.$$ Here, $A$, $B$, $C$ ...
0
votes
1answer
7 views

Surface area of a revolution integral problem

$y=x^3, 0 \leq x \leq 2 $ rotated around the x-axis. Here is my work, $$\begin{align} S &= \int_{0}^{2} 2\pi x^3 \sqrt{1 + (3x^2)^2}dx \\ &= 2\pi x^3 \sqrt{1+9x^4}dx \\ &= 2\pi \int_{0}^{...
2
votes
1answer
19 views

Finding a Probability for a Gamma Distributed Random Variable

Below is a problem from the book "Probability and Statistics". It is one of the Schaum's books. I am getting the wrong answer and I believe that I am doing the integration incorrectly. Thanks Bob ...
0
votes
0answers
30 views

If a continuous function $f: U \to \mathbb R$, $U$ open, has compact support, then f is Riemann integrable on $U$. What's $\tilde f$ an extension of?

My book is An Introduction to Manifolds by Loring W. Tu. My question is about a proposition in Section 23, which assumes a little familiarity with integration on $\mathbb R^n$ but gives an ...
0
votes
1answer
34 views

Integration intervals

I was doing an exercise about calculating the mass and centre of mass of a plane region defined by $P=\{(x,y): \vert y\vert \leq x \leq 1\}.$ I came across the integration interval for the integral ...
0
votes
1answer
42 views

A compactly supported continuous function on an open subset of $\mathbb R^n$ is Riemann integrable. What is the relevance of openness in the proof?

My book is An Introduction to Manifolds by Loring W. Tu. My question is about a proposition in Section 23, which assumes a little familiarity with integration on $\mathbb R^n$ but gives an ...
0
votes
1answer
31 views

Integration, Area under curve

The question goes by : Find the finite region bounded by the curve $$x=5t^2$$ $$y=2t^3$$ and the line $x = 5$. Find also the volume of the solid formed when this region is rotated through $\pi$ ...
0
votes
0answers
27 views

Integration of functions with integral

(1) $\int_{0}^{\infty} (\int_{t-x<v\le t, v>t-u} \frac{λ^k}{(k-1)!} v^{k-1} e^{-λv}dv)λe^{-λu}du$ (2) $\int_{0}^{\infty} (\int_{z\le t, t<z+u \le x+t} \frac{λ^k}{(k-1)!} z^{k-1} e^{-λz}dz)λe^...
2
votes
1answer
25 views

General Solution to Laplace's Equation in $\mathbb{R^3}$

I am trying to find the Green's function for $\ \nabla^2\phi=S(x)\ $ for $\ x\in\mathbb{R^3}$ and express the general solution to Laplace's equation in $\mathbb{R^3}$. To find the Green's function, I ...
1
vote
1answer
25 views

Martingale if and only if integral against continuous, bounded function is zero.

I have seen it written that the integrable process $(X_1,\ldots,X_n)$ is a martingale if and only if $$ \mathbb{E}[h(X_1,\ldots,X_k)(X_{k+1} - X_k)] =0, $$ for all continuous, bounded functions $h$, ...
0
votes
0answers
11 views

Integration involving x at the power and the gamma function

Suppose a nonnegative function $f$ on the positive real line and $f(x)\neq\Gamma(x)$. Do we know any closed formula of the following integral $$\int_{\mathbb{R}^+} \frac{\alpha^x}{\Gamma(x)}f(x)dx,$$ ...
0
votes
0answers
17 views

How to show the equation by integrating by parts .

Let $\Omega \subset \mathbb{R^n.}$ For $u \in C^1(\Omega) $ and $ \phi \in C_0^{\infty}(\Omega)$ one has $\int_{\Omega}\nabla|u|\phi dx=-\int_{\Omega}|u|\nabla\phi dx$
-2
votes
0answers
22 views
1
vote
1answer
27 views

Finding a subset $D$ such that the restriction of $f$ to $D$ is not Riemann-integrable

For $\mathbf{I}$ a generalized rectangle in $\mathbb{R}^{n}$, define $f : \mathbf{I} \rightarrow \mathbb{R}$ to be the function with constant value $1$. Find a subset $D$ of $\mathbf{I}$ such that ...
0
votes
0answers
17 views

Integration Proof with multiple variables

I just had a question regarding an exercise question that I received after today's lecture. I'm not too sure where to start. The question is seen in the link below where we are asked to: Show That: ...
2
votes
1answer
37 views

Change of limits in triple integral

The question is If $f$ is continuous, show that $$\int\limits_0^x\int\limits_0^y\int\limits_0^z f(t)dtdzdy = \frac{1}{2}\int\limits_0^x (x-t)^2f(t)dt.$$ The solution is \begin{equation*} \begin{...
0
votes
0answers
8 views

Double partial differentiation with respect to variables located at the integral limit

I have an integration over variable x, the limits of the integration are variables (z and y). I would like to make double partial differentiation for integral equation with respect to z and y. Is ...
1
vote
0answers
18 views

Remainder Estimate for Integral Test

enter image description here Hi i am unsure how to go about solving this problem. Could someone guide me through this?
3
votes
1answer
65 views

Trying to integrate $\iint_D \frac{\text{d}x\text{d}y}{x^2+y^2}$

I'm trying to integrate this: $$\iint_D \frac{\text{d}x\text{d}y}{x^2+y^2}$$ on $D=\{(x,y)\ |\ 1\le e^x+e^y, e^{2x}+e^{2y}\le 1\}$ After trying some substitution, I can't find a resonable way to ...
5
votes
1answer
55 views

Integral that is continuous and looks like it converges to a geometric series

I think the key word is continous. the RHS totally looks like a sum from a geometric series but I dont see a trick when I think there is one .
2
votes
2answers
33 views

Invert double integral

I'm trying to reverse the order of integration of the following double integral: $$\int _0^2\int _x^{2x}x^2dydx$$ I am aware that it is not possible to invert all double integrals, but my teacher told ...
0
votes
2answers
28 views

I need Help with Solving: $\int{y(t)*\ddot{y(t)}}dt$?

As I was solving a differential equation I came across this integral which I have no idea how to solve, I managed to reduce the equation to : $$Ct = y(t)\dot{y}(t)-y(t)$$ where $C$ is just a ...
2
votes
7answers
70 views

Evaluating $\int_{0}^{\frac{\pi}{4}}\frac{x\sin(x)}{\cos^{3}(x)}\,dx$

$$\int_{0}^{\frac{\pi}{4}}\frac{x\sin(x)}{\cos^{3}(x)}\,dx$$ I started like this: $$\int_{0}^{\frac{\pi}{4}}\dfrac{x\sin(x)}{\cos^{3}(x)}\,dx=\int_{0}^{\frac{\pi}{4}}\dfrac{1}{\cos^{2}(x)}\cdot \...
1
vote
1answer
27 views

Using knowledge of derivatives, what is f(x), when $f'(x)=\frac{1}{1+x^2} - sec^2x$ with initial condition f($0$)=$1$?

Second part of it is anti-derivitive of $\,\tan x\,$ I am having hard time with fraction in first part. I know it can written as $(1+x^2)^\frac{1}{2}$? Thank you!
0
votes
0answers
13 views

Integration minimum of two variables

I am trying to integrate the below equation: $$ \int_{\tau}^{\infty} \exp\left(-\frac{h}{\alpha} - \frac{\min\left(h,w\right)}{\beta}\right) \,\mathrm{d}h $$ where $\tau \leq h,w$ Using Indicator ...
0
votes
0answers
19 views

Solve contour integral using residue theorem [on hold]

I want to solve the following integral using the residue theorem, although I find it really tricky. Any ideas? $$ \int_C 1 / (z - i)^3 dz $$ Where $C$ is a circle with center $K(i)$ and $R=1$
2
votes
2answers
65 views

How do I integrate this equation with respect to $dcos\theta$?

I have this equation: $$\int_0^{-1}\big[(1+\cos^2\theta)+\cos\theta\big]d\cos\theta$$ I tried to do a u substitution. However, this didn't work. How can I evaluate this integral? Edit: Also ...
0
votes
1answer
19 views

Volume of a defined region using triple integrals

I was doing some excercises and I came upon this one, but I couldn't define the limits of integration. The problem says the following: Find the volume of the region defined by: $$z = x^2 + 3y^2 ~...
1
vote
1answer
23 views

Integral of Dirac-delta function from convolution theorem

In a question I have been lead to use the convolution theorem to find the inverse Laplace transform, as shown below: $$\omega(t)=\mathscr{L}^{-1}\left[e^{-bs}\frac{s}{s^2+a^2}\right]$$ From the ...
0
votes
0answers
35 views

How do I integrate this term?

Let $\Omega \subset \mathbb{R^n}$ , $u\in C^1(\Omega)$ , $\phi \in C^{\infty}_0(\Omega) $ a test function . $\int_{\Omega}\nabla |u| \phi dx=-\int_{\Omega}|u|\nabla\phi dx=-\int_{\{x\in\Omega|u>0\}...
0
votes
0answers
16 views

Integrating distance function over circle

I'm confused about the following integral. We fix $s\in S^1$: $$\int_{S^1}d_{S^1}(s,y)dH_{S^1}^{1}(y)$$ I tried to compute it brute force by separating the integral in two parts: $$\int_{S^1}d_{S^1}...
1
vote
1answer
27 views

Help integrating $\int_0^t\lambda e^{(a+b-1)\lambda x}(1-e^{-\lambda x})^{b-1}dx$

How to calculate the integral $$\int_0^t\lambda e^{(a+b-1)\lambda x}(1-e^{-\lambda x})^{b-1}dx~,$$ where all $\lambda,\;a$ and $b$ are constants? WolframAlpha won't solve it, but is there some kind of ...
1
vote
1answer
21 views

Find the area of the surface for the curve rotated about the x-axis

$y = \tan(x), 0\leq x \leq \frac{\pi}{3}$ I am struggling with constructing an integral for this formula. Since the curve is rotated about the x-axis, I think this is the best way to setup the ...
0
votes
1answer
31 views

Proof of the existence of the Ramanujan–Soldner constant

I know that the Ramanujan–Soldner constant is the positive zero of the Logarithmic integral defined as $li\left(x\right) = \displaystyle\int_{0}^{x} \frac{dt}{\log t}$ and that it is equal to: $\mu \...
1
vote
2answers
112 views

Integral $\int_{0}^{1} \int_{0}^{1} \frac{1}{(1+x y) \ln (x y)} d x d y$

Evaluate $$\int_{0}^{1} \int_{0}^{1} \frac{1}{(1+x y) \ln (x y)} d x d y$$ I couldn't get very far on this one, so I would appreciate some help =)
2
votes
2answers
110 views

Find : $\int_0^{\infty}\frac{\cos (2ax)}{x}\tanh (2πx)dx$

I'm try to Find : $$\int_0^{\infty}\frac{\cos (2ax)}{x}\tanh (2πx)dx$$ I don't have any idea to compute this type of integration Thanks!
0
votes
1answer
56 views

How to compute the limit of $\lim_{n\to \infty} \int_{[0,1]} n\frac{\sin x}{x} e^{-nx} \sin(nx) dx$

I tried to apply the Dominating Convergence Theorem but I can't find a function $g(x)$ such that $|f_n(x)| < g(x)$. Can you give me some hint for this problem? I tried to change the variable y = ...
0
votes
1answer
28 views

Integration of gamma function

Insurance company has to pay payments at the rate of $d$ per year. They are payable continuously as long as the person remains sick. The length of the payment period in years is a random variable ...
0
votes
0answers
26 views

imporper integral involving tanh and exponential. [on hold]

I'm stucking at calculate the integral of the form: $$\int_{-\infty}^{\infty}x\,e^{-a(x^2+b)}\,\tanh{(cx)}\,\mathrm{d}x$$ with $a,b,c>0$. Can someone offer a solution or reference? Thanks a lot.
2
votes
1answer
22 views

Computing $\mathcal{F}^{-1}\left(\frac{1}{(1+ik)(2+ik)}\right)$ via Convolution Theorem - Heaviside Step Function

I am trying to solve $$\mathcal{F}^{-1}\left(\frac{1}{(1+ik)(2+ik)}\right),$$ using the convolution theorem. While I am very aware that this problem is more easily solved using partial fractions, this ...
1
vote
0answers
22 views

How to show flow of vector field $(-f(y))$ is $\phi_{-t}(x)$ where $\phi_{t}(x)$ is the flow of $(f(y))$?

Let $f: E \rightarrow \mathbb{R}^n$ be a locally Lipschitz vector field for Initial Value Problem (IVP) $\dot{y} = f(y)$ with $y(0)=x$, where $E \subset \mathbb{R}^n$ is open. Definition: Let $\...
0
votes
0answers
16 views

Integrating a two Dimension function using one dimension delta

I know that given a function $\phi(x)$ and connected open interval $\Omega\in\mathbb{R}$ then \begin{align} \int_{\Omega}\phi(x)\delta(x-b)\;\mathrm{dx}=\phi(b) \end{align} I would like to know if $\...
1
vote
1answer
68 views

How to integrate $\arctan^2(x)$

I was wondering how you integrate $\arctan^2(x)$. I tried doing it by parts and allowing $u=\arctan(x)$ and $\frac{\mathrm dv}{\mathrm dx}=\arctan(x)$. But from then it becomes complicated, I was ...
0
votes
1answer
19 views

Applying Fourier transform to represent equation with integral as a sum of variables

According to a paper the equation with integral $\int_{-\infty}^{\infty}dx \rho_0(\lambda)e^{-b\lambda}=1/N$ (#1) where $\rho_0(\lambda)$ is a distribution function, $N$ is a natural number, can be ...
0
votes
2answers
31 views

Find the exact area of the surface obtained by rotating the curve $x=1+2y^2$ about the x-axis.

Currently I am studying how to integrate the area of a surface of revolution. $$x = 1+2y^2,~~1\leq y\leq2 \textrm{ around the x axis}$$ Rewrite function in terms of x and find the derivative of $f(...
0
votes
0answers
33 views

Derivative of integrals with respect to a function inside the integral

I have a maximization problem that requires the derivative of: $\int_0^n g(x)^\rho dx$ and $\int_0^n g(x)p(x) dx$ with respect to $g$ I've used the chain rule to substitute $dx = dg(x)(g'(x))^{-1}$ ...
1
vote
0answers
42 views

Integral with Logs

I have a two integrals mathematica won't solve and I'm not sure how to solve them although I'm certain they are solvable. The first one is: $\int_{0}^\pi \log\left(A-B\cos\theta\right)\left(A+B\cos\...
1
vote
1answer
24 views

How to know whether to choose the x-bound or the y-bound for this triple integral

In my textbook for calculus 3, I have been working on example of the triple integral. Though I do know polar, cylindrical, spherical coordinates, this section of the book expects you to work with ...