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
19
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9
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Formula for bump function
I would like to formulate a bump function (link) $f(x)$ with the following properties on the reals:
$$
f(x) :=
\begin{cases}
0, & \mbox{if } x \le -1 \\
1, & \mbox{if } x = 0 \\
0, & \...
- 348k
-1
votes
0
answers
27
views
How to compute Contour Integral Numerically
So I am trying to compute the integral from $0$ to $a$ on the real axis, shown in the picture, for a function that is completely analytic on the upper half plane except at $E + i\epsilon$ where $\...
- 9
105
votes
7
answers
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views
Does L'Hôpital's work the other way?
As referred in Wikipedia (see the specified criteria there), L'Hôpital's rule says,
$$
\lim_{x\to c}\frac{f(x)}{g(x)}=\lim_{x\to c}\frac{f'(x)}{g'(x)}
$$
As
$$
\lim_{x\to c}\frac{f'(x)}{g'(x)}=
\lim_{...
- 167
8
votes
4
answers
331
views
Evaluation of $\int\frac{1}{x+ \sqrt{x^2-x+1}}dx$
Evaluate :
$$\int\frac{1}{x+ \sqrt{x^2-x+1}}dx$$
After multiplying the denominator with $x-\sqrt{x^2-x+1}$, I get
$$x+\ln |x-1|-\int \frac{\sqrt{x^2-x+1}}{x-1}dx$$
Is $\int \frac{\sqrt{x^2-x+1}}{x-1}...
- 1,011
0
votes
1
answer
43
views
How do I approach this volume problem?
I'm trying to find the volume of a body bounded by the surface given by the equation $(x^2 + y^2 + z^2)^2 = (x^2 - y^2)z$, where $z \geqslant 0$. Here's my thought process: by observing that the we ...
- 8,569
2
votes
1
answer
90
views
M is composed of line segments connecting ellipse to $(0,0,0)$ Calculate integral $\int_M \sqrt{x + 3z}\ d \lambda_2$ over those. Almost done.
I found such an exercise among my set of exercises preparing for exams and I have no idea how to solve that.
Every point of ellipse $\{ (x,y,z) \in \mathbb{R}^3 : x^2 + y^2 = 1, x + z = 1 \}$ is ...
- 1,096
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.
$$\int_0^1\int_0^1\frac{\partial^2}{\partial\rho^2}\{s\rho^2C(\textbf{r}_2,ss^\prime\rho)...
- 140
0
votes
0
answers
78
views
Looking for citation of a definition of multivariable integral
I need a definition of Multivariable Riemann Integral to cite in my article.
I've been searching different cited sources, for example Rudin, Folland, Stewart and Larson books.
In the first two I ...
4
votes
2
answers
309
views
Showing $\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right) \, dx=\frac{\pi^2 + 4}{2}$
While exploring possible applications for exponential substitution, I stumbled upon the following integral identity:
$$\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right)...
- 1,011
2
votes
1
answer
85
views
How to Solve $\int \frac{x + 1}{(x^2 + 6x + 14)^3} dx$? [duplicate]
I am trying to solve the following integral and would appreciate some guidance:
$$\int \frac{x + 1}{(x^2 + 6x + 14)^3} \, dx$$
I have attempted various methods, including substitution and integration ...
- 3,738
1
vote
0
answers
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views
Excess part from integration of graph
Photo of Graph of a arbitrary function on which integration is performed
Here, in the photo, using the basic idea of integration, we make a long rectangular block to cover the area under the graph. ...
- 11
1
vote
1
answer
35
views
Double integral of $\iint x\,\mathrm dx\,\mathrm dy$ using polar coordinates
Solve$\newcommand{\dd}{\mathrm d}$
$$\iint_D x\,\dd x\,\dd y$$
using $x= 2r \cos \phi, y= r \sin \phi$ over D: $x^2+4y^2 \leq 4, x \geq 0$.
I obtain $r \leq 1$ and $\frac{3\pi}{2} \leq \phi \leq \frac{...
- 7,184
0
votes
0
answers
82
views
Understanding of the term “compact support” in the proof of stokes theorem.
I am trying to understand the proof of Stokes theorem:
$$
\int_M df = \int_{\partial M} f
$$
for a differentiable Manifold $M$ with dimension $n$ and a differential $(n-1)$-form with compact support ...
- 101
-1
votes
2
answers
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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 $2x + 2y > 0 \iff x + y > 0$
...
- 23.6k
2
votes
2
answers
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How to modify Gauss-Hermite quadrature rule when the weight function is slightly generalized
I hope this is the right forum.
Consider a slightly modified version of the Gauss-Hermite Quadrature Rule, where the weight function is not $\exp\left(-x^{2}/2\...
- 91k