All Questions
Tagged with integration definite-integrals
13,448
questions
1
vote
2
answers
71
views
Complex integrals that look like they agree, differ by sign (according to Mathematica)
Consider the integral
$$\int_0^\infty \frac{dz}{1-z^2 +i0^+},$$
I would assume it to agree with the integral
$$\int_0^\infty \frac{dz}{(1-z+i0^+)(1+z+i0^+)}. $$
However, according to Mathematica the ...
0
votes
0
answers
67
views
Double integral $ \iint_D (x^4-y^4) dx\,dy$
I have troubles with the following integral
$$
\iint_D (x^4-y^4) dx\,dy
$$
over D: $1<x^2-y^2<4, \sqrt{17}<x^2+y^2<5, x<0, y>0$
This is the same problem as in Compute $\iint_D (x^4-y^...
0
votes
0
answers
35
views
Moments of Pearcy type integral
In my research I encounter Moments of Pearcy Integral
which can be written as
$$
\int_{-\infty}^{\infty}x^{n}
{\rm e}^{-ax^{4} + bx^{2} + cx}\,{\rm d}x\qquad
a > ...
- 41
0
votes
0
answers
50
views
Approximation of a Riemann sum.
Given a twice continuously differentiable function $f\in C^2([0,1])$, is there a theorem/result/algorithm on how to place $0<x_1<\ldots<x_{n-1}<1$ so that adding $x_0=0$ and $x_n=1$,
$$
\...
- 208
-1
votes
2
answers
62
views
$A = \{ x^2 + y^2 + z^2 < 2x + 2y \} \subset \mathbb{R}^3$.Calculate $\int_A xyz \ d \lambda_3$. I need to verify my solution.
$A = \{ x^2 + y^2 + z^2 < 2x + 2y \} \subset \mathbb{R}^3$
Calculate:
$$\int_A xyz \ d \lambda_3$$
Solution:
We know that: $x^2 + y^2 + z^2 > 0$ and therefore $2x + 2y > 0 \iff x + y > 0$
...
- 1,096
0
votes
0
answers
25
views
Will the following Method of engineering analysis work?
Analytical Engineering Analysis of 3D Shapes
Using volume integral($\iiint_{}^{}{f(t)}dx dy dz$) to do a Analytical
Engineering Analysis of 3D Shapes without using mesh based FEA. Like
integration ...
19
votes
9
answers
2k
views
Formula for bump function
I would like to formulate a bump function (link) $f(x)$ with the following properties on the reals:
$$
f(x) :=
\begin{cases}
0, & \mbox{if } x \le -1 \\
1, & \mbox{if } x = 0 \\
0, & \...
2
votes
1
answer
208
views
How to evaluate $\int_0^{\infty } \frac{\sin (\pi x)}{\log (x)} \, dx$
I have tried different substitutions and transformations, but am not getting a lead. Any suggestion would be helpful.
The numerical value of the integral is around $-3.2192$.
Interestingly these two ...
0
votes
2
answers
114
views
Variable transformation in the definite integral
recently I encounter a variable transformation problem in the derivation and I did not figure out how it works.
$$\int_0^1\int_0^1\frac{\partial^2}{\partial\rho^2}\{s\rho^2C(\textbf{r}_2,ss^\prime\rho)...
- 41
0
votes
2
answers
53
views
Integral of a Generalized Laguerre Polynomial [closed]
I am looking for the solutions to the following integral:
$$
I_{n} =
\int_{0}^{\infty}x^{4}
\operatorname{L}_{n}^{3}\left(x\right)
{\rm e}^{-\left(n + 3\right)x/2}\,{\rm d}x,\qquad n \in\mathbb{N}_{0}
...
- 19
2
votes
1
answer
76
views
Double integral of $x^2y+y \sin(x^9)$ dxdy
I have some troubles with the following double integral (in particular the part with sinus)
$$
\iint_{D}\left[x^{2}y + y\sin\left(x^{9}\right)\right]{\rm d}x\,{\rm d}y\quad
\mbox{where}\quad D\ \mbox{...
2
votes
2
answers
149
views
Evaluating $I=\int_1^\infty \frac{2(z^2-1)}{(z^2+1)^2\ln(z)}\,dz$
$$
I =
\int_{0}^{\infty} \frac{\tanh\left(x\right)\operatorname{sech}\left(x\right)}{x}\,{\rm d}x
$$
I started with the substitution $\cosh x=t \implies [0,\infty] \to[1,\infty]$
$$I=\int_1^\infty \...
- 2,879
0
votes
2
answers
85
views
Area with double integral in polar coordinates
Determine the area interior to $y^2=2ax-x^2$ and exterior to $y^2=ax$.
The area in artesian coordinates is $$\int_{0}^{a}\int_{\sqrt{ax}}^{\sqrt{2ax-x^2}} dydx$$. To convert it into polar coordinates ...
- 689
0
votes
1
answer
57
views
Double integral of the form exp(-a(x-y)^2) [closed]
I would like to compute the value of the double integral :
$$\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
-2
votes
1
answer
46
views
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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