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.
60,099
questions
0
votes
0answers
8 views
compute $\iint_Y F.N \ dS$
The question is: Find $\iint_Y F.N \ dS $
$$ F=(x^4+yz-x^5,5x^4y,z),\quad \text{The surface }Y=x^2+y^2-z^2=1, \ \ 0\leq z\leq 1 \quad N=\text{
the normal points away from z axis}$$
Here is how i ...
0
votes
0answers
11 views
Question on marginal probability density function
Consider $\zeta \sim U[-2; 2]$, $\eta \sim U[0; 1]$, $Z = \zeta + \eta\zeta$, $\zeta$ and $\eta$ are independent.
First of all, I need to find the conditional density of $Z \vert\zeta=x$. Let $\zeta = ...
0
votes
1answer
34 views
Finding $\int_{1}^{2} \sqrt{1+\frac{1}{x^2}+\frac{1}{(x+1)^2}} dx$
This integral $$\int_{1}^{2} \sqrt{1+\frac{1}{x^2}+\frac{1}{(x+1)^2}} dx$$
has been fabricated keeping an interesting point in mind, I may present a solution later. The question is: How will you do it ...
1
vote
2answers
36 views
Integrate $\int \frac{\tan \left(x\right)}{\sin ^2\left(x\right)}dx\cdot \int \log _{3x}\left(x^2\right)dx$.
Integrate the following integral:
$$\int \frac{\tan \left(x\right)}{\sin ^2\left(x\right)}dx\cdot \int \log _{3x}\left(x^2\right)dx$$
I did this question a while ago and the answer is correct. ...
0
votes
2answers
22 views
Evaluate the integral using the given data
Let $f:[0,1]\to [0,1]$ be a continuous function such that $x^2 +(f(x))^2\le 1$ for all $x\in [0,1]$ and $\int_0^1 f(x).dx=\frac{\pi}{4}$, then find $\int_{-1/2}^{1/\sqrt 2} \frac{f(x)}{1-x^2}.dx$
The ...
0
votes
2answers
28 views
How to find the volume of this region?
I want to find the volume of the following subset of $\mathbb R^3$:
\begin{equation}
(x^2+y^2)^2\leq x,\quad 0\leq z\leq 2x-\sqrt{x^2+y^2}\text{.}
\end{equation}
I tried to draw a picture of the given ...
2
votes
1answer
49 views
Number of roots of the equation $f(x)= \int_0^x (t-1)(t-2)(t-3)(t-4)dt =0$
I have the following question before me:
Find the number of roots of the equation $f(x)= \int_0^x (t-1)(t-2)(t-3)(t-4)dt =0$ in the interval $[0,5]$.
$0$ is clearly one of the roots. But how can I ...
0
votes
0answers
27 views
Integration by subsitution
Im confused by the subsitution, I would've assumed it would be
I can give more context if needed, its from the paper. On the generality of the relationship among contact stiffness,
contact area, and ...
1
vote
1answer
47 views
Integral trick explanation or link
We used the following integral trick in our lecture, I really can't understand why it works. I would appreciate an explanation or even just a name so i can search for it :)
So apparently:
$$\int_{-\...
0
votes
2answers
24 views
Calculate the following integral in the given set
I have to calculate the Integral of the following function in the given set D.
Now I went on to try to write this D as a set of the form $ E$ = {$(x,y) \in \mathbb{R^2}: a < x < b , \alpha(x)&...
0
votes
0answers
10 views
Pointwise convergence and derivatives [duplicate]
Let $f,g$ $C^0$ functions from $[0, 1] \to \mathbb R$, and a sequence of $C^1$ functions $f_n : [0, 1] \to \mathbb R$. Show that if $f_n'$ converges uniformly to $g$ and $f_n$ converges pointwise to $...
0
votes
1answer
36 views
Problem with Poisson integral
I have heat equation
$$
u_t=u_xx \\
u(x,0)=xe^{\frac{-x^2}{3}}
$$
I solve by the Poisson integral
$$
u(x,t)=\frac{1}{2\sqrt{\pi t}} \int xe^{\frac{-x^2}{3}}e^{\frac{-(y-x)^2}{4t}} dx
$$
I worked with ...
0
votes
0answers
13 views
Find the relationship between $x$ and $y$ so that $y:=0\rightarrow\frac{\pi}{2}\Leftrightarrow x:=y\rightarrow\frac{\pi}{2}.$
Find the relationship between $x$ and $y$ so that $y:=0\rightarrow \dfrac{\pi}{2}\Leftrightarrow x:=y\rightarrow \dfrac{\pi}{2}.$
I'm having trouble solving the double integral if I change the order ...
4
votes
1answer
28 views
Finding the Riemann $\zeta$ function by adelic integration
I am referring to Tao's blog post about Tate's thesis. Introduce the adeles $\mathbb A$ of $\mathbb Q$ and the adelic Mellin transform
$$Z(s) = \int_{\mathbb A^\times} = g(x) |x|^s d^\times x.$$
Here, ...
3
votes
1answer
26 views
Find $f:[0, 1]\to \mathbb R$ that maximizes $I(f)-J(f)$, where $I(f)=\int_0^1 {x^2 f(x)dx}$, $J(f)=\int_0^1{x\left(f(x)\right)^2 dx}$
I've never seen this kind of problems - finding a function with almost no conditions which maximizes the integral - so I'm asking for a hint. The problem is as follows.
Find a continuous function $f:[...
0
votes
0answers
33 views
Improper integrals ; Why does this equality hold?
$
H(t):=
\begin{cases}
\dfrac{1}{2}-t & (0<t<1) \\
0 & (t=0) \\
\text{periodic of period 1} &(\text{otherwise})
\end{cases}$
$
\mu (x):=\sum_{n=0}^{\infty} \displaystyle\int_0^1 \...
1
vote
1answer
37 views
A p-adic Fourier transform
Consider the field of $p$-adic numbers $\mathbb Q_p$. Define the character $\chi(u p^n) = e(p^n)$ for all $n \in \mathbb Z$ and all unit $u$. In particular it is trivial on integers. This allows to ...
0
votes
1answer
84 views
Differentiate under the integral sign: $ \int_0^{2Ļ} e^{\cos (x)}\cos(\sin x) \, \mathrm{d}x $
I have again a doubt regarding an exercise of differentiation under the integral sign. In this case, it concerns the integral:
$$ \int_0^{2Ļ} e^{\cos(x)}\cos(\sin x) \, \mathrm{d}x $$
I tried the ...
0
votes
1answer
37 views
Theorem 6.10 Rudin PMA, Partition
The third paragraph of the proof begins,
'Now form a partition $P = \{x_0,x_1,...,x_n\}$ of $[a,b]$, as follows: Each $u_j$ occurs in $P$. Each $v_j$ occurs in $P$. No point of any segment $(u_j,v_j)$ ...
0
votes
1answer
46 views
Line integral calculation for a line segment
Compute the line integral $\int_y e^z dz$ where $y$ is the line segment from $0$ to $z_0$.
Since there is only one variable z here, I can directly compute $\int_y e^z dz=\int_0^{z_0} e^z dz=e^{z_0}-1$...
0
votes
1answer
32 views
Standard path for integral with complex limits?
my question is quite simple and might be a duplicate though I couldn't find one. Is there an accepted meaning for the notation
$$\int_{z_1}^{z_2}$$
With $z_1,z_2\in\mathbb{C}$? Is the standard to just ...
1
vote
1answer
54 views
Particular integral for $a\frac{d^2y}{dx^2}+b\frac{dy}{dx}+cy=\ln x$
I would like to know how to find a particular integral for $$a\frac{d^2y}{dx^2}+b\frac{dy}{dx}+cy=\ln x$$
where $a,b,c$ are constants. So far, the only functions I've come across for $f(x)$ in
$$a\...
1
vote
1answer
31 views
How to calculate this integral: $\int_{C}{} (z^2-2z) dz$ where $C$ is a line segment starting at $1$ and ending in $i$
The task is as follows:
$$
\text{Calculate } \int_{C}{}f(z) dz \text{, where }
\\
\text{a) } f(z)=z^2-2z \text{, } C \text{ - line segment starting at } 1 \text{ and ending at } i \text{;}
\\
\text{b) ...
0
votes
1answer
25 views
Volume using triple integral with spherical and cylindrical coordinates [closed]
I had to solve this triple integral and I tried to solve by cylindrical and spherical coordinates but couldn't get anywhere. I was hoping someone could help me in this problem.
Solve $\int \int \int _{...
2
votes
0answers
48 views
$\lim_{x\to a}\frac{a}{x-a}\operatorname{Si}_a(x)$
Let $a>0$ and $\operatorname{Si}_a(x)=\int_{a}^{x}\frac{\sin(t)}{t} \, dt$. Compute
\begin{equation*}
\lim_{x\to a}\frac{a}{x-a}\operatorname{Si}_a(x)
\end{equation*}
My reasoning was: suppose $F$ ...
0
votes
0answers
16 views
Fourier series transformation help - unable to reconstruct (simple) paper results
I am reading a paper and am having difficult reconstructing an equation.
The paper begins with the following equations where $\omega_{n} = \omega_{o} + n\Delta\omega$ and:
(1) $A_{in}(t) = \sum_{n}a_{...
1
vote
4answers
38 views
Solid of revolution axis $y=5$
the problem goes like this
Rotate the indicated area around the given axis to calculate the volume of the solid of revolution
$y=x^2+1$, $x=0$, $x=2$, $y=0$ , around the axis $ y = 5$
My question is ...
1
vote
0answers
34 views
How do you find the complex $(C_n)$ fourier series expansion of $e^{ax}$?
How do you find the complex $(C_n)$ fourier series expansion of $e^{ax}$?
The period of the function is 2$\pi$.
I've tried to do this question out several times, and I keep getting stuck and don't ...
0
votes
0answers
21 views
Variable change in multiple integrals: polar coordinates at zero.
We can make a variable change if we have some diffeomorphism $\phi$. A usual variable change is polar coordinates. But when $\rho = 0\quad |\phi'| = 0$, so $\phi$ is not a diffeomorphism if our region ...
4
votes
4answers
128 views
Evaluating $\int_{0}^{\pi} \frac{1}{\sqrt{u^2+2u\cos x +1}} \mathrm{d}x$
Does the integral $$\int_0^{\pi} \frac{1}{\sqrt{u^2+2u\cos x +1}} \text d x \hspace{30pt} (u \le 1) $$ has a closed form? If it has, how do we evaluate it?
I was solving a physics problem which I ...
-4
votes
1answer
60 views
Challenging integration problem of an exponential function
While preparing a tutorial session for my students, I come across a very challenging integration, which is :$$ \int_{-\infty}^\infty e^{t^2 }\ \mathrm{d}t$$
I attempted many methods to solve it but ...
1
vote
0answers
92 views
How to solve this triple intergral $I=\iiint_{Q}\frac{1}{x^2+y^2}dv$?
$$
I \equiv \iiint_{Q}\frac{{\rm d}v}{x^2+y^2}
$$ Which $Q$ is a solid bounded above by $z=4-x^2-y^2$ and below by the sphere $x^2+y^2+z^2=9$.
I have tried with this multiple integration by using ...
1
vote
1answer
33 views
Show that $L(f)=\sup\{L(P,f):P \in \mathcal P_{c}\}$,where $c\in \left(a,b\right)$
Assume $\mathcal P_{[a,b]}$ is the set of all partitions of $[a,b]$ and $\mathcal P_{c}$ is the set of all partitions of $[a,b]$ containing $c$,where $c\in \left(a,b\right)$,if $f:I \to \mathbb R$ ...
1
vote
0answers
19 views
KL divergence between 2 joint probability distribution
Consider the following setting :
$$Q(x,z)= q_{data}(x)q_{\phi}(z|x) \;\;\;\text{and} \;\;\; P(x,z)= p(z)p_{\theta}(x|z) = p_{\theta}(x)p_{\theta}(z|x)$$
I want to compute $KL(Q||P)$.
Here is how I am ...
1
vote
1answer
38 views
Show that $m\le\left(\frac{1}{b-a}\int_{a}^{b}f^{2}\left(x\right)dx\right)^{\frac{1}{2}}\le M$
Assume $f$ is a real-valued function which is integrable over the interval $I=[a,b]$ and for every $x \in [a,b]$ we have that $0\le m \le f(x) \le M$,show the following inequality does hold:
$$m\le\...
0
votes
0answers
8 views
Find isovalue such that a volume integral under the isosurface has a specific value
Lets say i have a continiously differentialbe function $f(x, y, z)$ that is nonnegative everywhere and has a finite volume integral over all space that yields 1,
$$
\int^\infty_{-\infty }dx\int^\...
0
votes
1answer
45 views
$\int \frac{1}{x(x+1)…(x+n)}dx$ [duplicate]
I have not managed to evaluate the integral. I feel the idea is just split the integral into a sum of integrals
$$\int \frac{1}{x(x+1)...(x+n)}dx = \int \left(\frac{A_1}{x} + \frac{A_2}{x+1} +... +\...
0
votes
1answer
34 views
Proof of generalized open & closed Newton-Cotes formulas
I am looking for proof of generalized open & closed Newton-Cotes formulas. I couldn't find any reference which properly proves both theorems. Most books just state the theorem and do not provide ...
0
votes
1answer
65 views
$I = \int_0^\infty y^2 [ (y^2 +1)/\sqrt{y^4 + 2 y^2 } - 1] dy$
I tried with contour integral:
$$I = \frac{1}{2}\int_{-\infty}^{\infty } dz z^2\left(\dfrac{z^2 +1}{\sqrt{z^4 + 2 z^2 }} - 1\right)$$
The contour can be deformed into the upper half plane. But there ...
1
vote
0answers
33 views
Integral of a measurable not non negative function over N
I know that given a non-negative function $f: \mathbb{N} \rightarrow \mathbb{R}$, the integral over $\mathbb{N}$ with the counting measure is just the infinite sum.
However, given the funcion $b(n) = \...
5
votes
0answers
73 views
Finding convolution of two functions
A common engineering notational convention is: wikipedia
${\displaystyle f(x)*g(x)\,:=\underbrace {\int_{0}^{x}f(\tau )g(x-\tau )\,d\tau } _{(f*g)(x)}.}$
I want to write the following expression as ...
0
votes
1answer
39 views
$\int_{0}^{\infty } e^{-t}\cdot t^{3}\cdot \sin(t) dt$ using Laplace transform
I am currently on this question:
Use Laplace Transform to evaluate the following
integral:
$\int_{0}^{\infty } e^{-t}\cdot t^{3}\cdot \sin(t) dt$
What I did:
let the $\int_{0}^{\infty } e^{-t } dt$ = ...
0
votes
1answer
52 views
Given a relation between $G$ and $g$ prove that $g(x)=x+1$.
I am having several difficulties in solving an exercise that has been put to me.
The exercise says:
If $g: (-1; \infty) \rightarrow \mathbb{R}^+$ is a function that can be antiderivated, let $G$ be ...
2
votes
0answers
83 views
Gauss Integration of $\sqrt{x}$
$w(x) = x^{1/2}$
$$\int_{0}^{1}x^{1/2}f(x)dx = a_{1}f(x_{1})+a_{2}f(x_{2})$$
$$\Pi_{2}(x^2 -p_{1}x+p_{2})$$
\begin{equation}
\int_{0}^{1}x^{1/2} \cdot \Pi_{2}(x) dx = 0
\end{equation}
\begin{equation}...
3
votes
0answers
36 views
Inverse Laplace Transform hint
I am away from Laplace transform for years, and now I have to solve $$\mathcal{L}^{-1} \left\{ s^{-\frac32}\sqrt{\frac{as+b}{cs+b}} \right\}$$a,b,c are real positive numbers.I can find inverse Laplace ...
0
votes
1answer
67 views
What does this definite integral theorem mean?
As I was solving a homework problem about definite integrals, I came across a theorem to help me solve the problem. Although I got the right answer to the problem, I do not really understand the ...
-1
votes
1answer
27 views
On the calculation of curve integral
Let $\vec{n}$ be the out of unit normal vector on the curve $\varGamma$,and the $\varGamma:x^2+y^2=R^2$.define$$r=\sqrt{x^2+y^2}.$$Calculate$$\oint_{\varGamma}\dfrac{\partial\ln r}{\partial{\vec{n}}}\...
-3
votes
0answers
21 views
Use the method of cylindrical shells to find the volume
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves x=sqrt(y), x=0 and y=8 about the x-axis.
1
vote
0answers
9 views
Implementing discrete derivative by integration over a set of data
I'm trying to implement Lanczos derivative $f'_L$ as an option to calculate over a set of data. Since i needed a discrete version of Lanczos derivative $f'_{DL}$, i rearranged the equation such as $$f'...
1
vote
1answer
23 views
Volume of a solid of revolution - Getting two different results.
The region between $y=x^2+1$ and $y=-x+3$ is rotated about the $x$-axis.
I have to compute the volume. The intersection between these two curves are at $x=-2$ and $x=1$.
At first, I thought about $V(x)...
