Questions tagged [integration]
For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.
74,716
questions
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A problem on dirt displacement
Definition. Given a function $f\in L^1(\mathbb{R})$ such that $xf\in L^1(\mathbb{R})$, the quantity $\int_\mathbb{R}xf(x)\,dx$ is called the unnormalized center of mass of $f$ and is denoted $UCM(f)$.
...
- 984
1
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1
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53
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Conditional expectation - alternative expression
Consider the following set-up.
$F:[0,\omega]\rightarrow[0,1]$ where $X$ is a real-valued random variable.
The conditional expectation of $X$ given $X<x$ is:
$E(X|X<x)=\frac{1}{F(x)} \int_0^s tf(...
- 1,467
-2
votes
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answers
100
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Evaluate $\int \frac{\ln(x+1)}{ \ln(x)}$ [duplicate]
Evaluate the integral: $\int \frac {\ln \left( x + 1 \right)}{\ln \left( x \right)} \text {d} x.$
I tried looking it up on Wolfram Alpha, but it immediately said its computation time was exceeded. I ...
0
votes
1
answer
121
views
Double integral of $xe^{-(x^2+y^2)}$
I have some troubles with the following double integral where D is $|x|\leq 1, |y|\leq 1$
$$
\iint_{D} xe^{-(x^2+y^2)} \,dx\,dy
$$
I transform it to polar coordinates where $\theta~is [0,\pi /2]:$
$$
\...
0
votes
2
answers
85
views
Area with double integral in polar coordinates
Determine the area interior to $y^2=2ax-x^2$ and exterior to $y^2=ax$.
The area in artesian coordinates is $$\int_{0}^{a}\int_{\sqrt{ax}}^{\sqrt{2ax-x^2}} dydx$$. To convert it into polar coordinates ...
- 689
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votes
1
answer
57
views
Double integral of the form exp(-a(x-y)^2) [closed]
I would like to compute the value of the double integral :
$$\int_0^1 \int_0^1 e^{-\gamma^2(y-x)^2} dx dy $$
where $\gamma \in \mathbb{R}$.
I think maybe we can do a change of variable in order to get ...
- 13
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0
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37
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Writing $\ln(x+1)e^{-ax}$ in terms of Meijer-G function
Is there any way to write $f(x)=\ln(x+1)e^{-ax}$ in terms of Meijer-G function? I tried calculating Mellin transform of $f(x)$ to no avail. Frustrated, I used Mathematica to get the following answer
$$...
- 3,263
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votes
0
answers
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views
Need help with the steps and limits in this multivariable integration of a joint probability density function
I am not sure how to proceed with this double integration. I know this can be evaluated in a much easier way than solving the integral as its just the volume of a cube but I need help with the process ...
0
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37
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Intergrating $ \int_{M} f (x.,y,z,w)\ d {\rm Vol}_3 $
I want to integrate $$ I = \int_{M} f (x.,y,z,w)\ d {\rm
Vol}_3 $$where $f(x,y,z,w) = (x+y)e^{z+w} $, and
$M = \{x+y+z+w = 1, x,y,z,w > 0\} $.
I need to find a parameterization of M; if I consider $...
- 391
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0
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44
views
The moment of multivariate normal distribution
This is a computational problem I ran into while reading an article. I describe my question below:
Let $\boldsymbol{Z}\sim N(0,I_{p\times p})$ and $\boldsymbol{y}_{i}\in \mathbb{R}^{p}$.
We need to ...
- 1
-2
votes
1
answer
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If $f$ and $g$ coincide almost everywhere on $[a, b]$, then is $\int_a^b f(x) dx = \int_a^b g(x) dx$? [duplicate]
Let $a$ and $b$ be any real numbers such that $a < b$, and let $S$ be a (nonempty) subset of the closed bounded interval $[a, b]$ such that $S$ has measure $0$. Now let $f \colon [a, b] \...
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What is the fault in this method of finding second moment of area of a circle
I am trying to find the second moment of area of a circle about a diameter using first principles.
Place the centre of the circle at the origin of XY-plane. Now consider a tiny circular sector with an ...
- 141
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$f(x) = k(x-a)(x-b)(x-c)(x-d)(x-e)(x-t)$ [closed]
Question - $f(x) = k(x-a)(x-b)(x-c)(x-d)(x-e)(x-t)$.
$f^{\prime}(x) = 6(x^5-8x^4+24x^3+Ax^2+Bx+c)$ if $ e\leq k_1 $ and $t \geq k_2$. $a<b<c<d<e<t$.
Find $k_1+k_2$.
My attempt - I ...
- 11
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3
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246
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Question regarding integral involving logarithm and sine [duplicate]
I have to compute the following integral
$$\int_{0}^{\pi/2} \frac{\ln(1-\sin x)}{\sin x} dx$$
I decided to solve this using the Feynman's Trick for integration and parametrized the integral as follows
...
- 356
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votes
0
answers
41
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Calculate the Following Complex Integral Around the Given Contours [closed]
$$
\oint_{\lambda} \frac{\cos^5(z)}{(z - i\pi)^3} \, dz
$$
$$
\|z\| = 1 \\
$$
$$
\|z + 1\| = \frac{\pi}2
$$
I have a question regarding the calculation of the integral. When using the residue theorem ...
