All Questions
522
questions
1
vote
1
answer
283
views
Proving a lower bound for this double integral
Let $f:\mathbb R^n\to\mathbb R$ be a smooth function. Let $k>0$ and consider the cutoff function $T_k:\mathbb R\to\mathbb R$ defined for any $t\in\mathbb R$ as
$$T_k(t)= \int_0^t 1_{\mathbb R\...
- 3,277
0
votes
2
answers
66
views
Is $\int_D \nabla f dx = 0$ for $f$ compactly supported?
Let $D \subset \mathbb{R}^n$ be bounded and $f$ a smooth compactly supported function such that its support is contained within $D$. I am interested in
$$\int_D \nabla f dx.$$
If $n = 1$, then by the ...
- 6,277
3
votes
0
answers
57
views
Calculation of the volume of a solid of rotation
Calculate the centroid of the homogeneous solid generated by the rotation around the y-axis of the domain in the xy-plane defined by: $$D=\left \{(x,y)\in \Bbb R^2 : x \in [1,2],0\leq y \leq \frac{1}{...
3
votes
1
answer
115
views
Computing an integral using differential under the integral sign
The following integral is in question.
$$I(x) =\int_0^x \frac{\ln(1+tx)}{1+t^2}\,dt$$
My attempt is finding $I’(x)$ which is
$$I’(x) = \int_0^x \frac{t}{(1+t^2)(1+tx)}\,dt + \frac{\ln(1+x^2)}{1+x^2} $$...
- 617
0
votes
1
answer
36
views
Parameterising Surfaces Integration
$S$ is the surface of a cube which is bounded by $6$ planes being $x=1,x=3,y=2,y=4,z=0,z=2$. The normal vector points outwards and the vector field is $F = (x^2-sin(yz),\frac{cos(x)}{x^2}-yz,x^2y)$
...
- 261
1
vote
1
answer
27
views
Flux through a paraboloid in the first quadrant
The question is absolutely easy but I am unsure where I go wrong haha. I am given a vector field $\mathbf{u} = (y,z,x)$ and I need to find the ourward flux $\iint\limits_{M} \textbf{u} \cdot d\textbf{...
- 617
0
votes
1
answer
40
views
Regions whose area cannot be measure in the sense of Riemann
Consider a region $D$ of the plane. Usually, its area is defined as
$$
\iint_D 1
$$
Would it be possible that this integral does not exists, and thus $D$ has a non-measurable area?
- 284
2
votes
2
answers
63
views
Multiple Integral Problem with Dirac Delta Constraint: Seeking Guidance
I am working on a challenging multiple integral problem and would appreciate any assistance. The integral is as follows:
$$
\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \ldots \int_{-\infty}^{+\...
1
vote
1
answer
61
views
Integrate a sum of trig function under absolute value
Let $n \in \mathbb{N}$, I'm trying to compute an explicit formula for the following integral:
$$
\operatorname{I}\left(n\right) = \int_{\left[0,2\pi\right]^{\,\,n}}\,
\left\vert\rule{0pt}{4mm}\,{\cos\...
- 888
0
votes
1
answer
49
views
Deal with discontinuity in double integrals
Let $f(x,y)= \frac{x^2+y^2}{x^2}$. Consider
$$ \int_D f(x,y)$$
with $D=\{ (x,y): 0\leq y\leq x, x^2+y^2\leq 1 \}$
The function is unbounded in $D$, due to the denominator. I checked that cannot be ...
- 499
1
vote
2
answers
133
views
Generalised integral $\int_{0}^{\infty} \int_{0}^{y} \sqrt{x^2 + y^2} e^{-x^2 - y^2} dx dy$
I am having trouble understanding how to compute this integral $$\int_{0}^{\infty} \int_{0}^{y} \sqrt{x^2 + y^2} e^{-x^2 - y^2} \, dx \, dy$$
My idea is to consider the whole plane xy plane instead of ...
- 617
0
votes
1
answer
34
views
Measure Theory: Proof regarding a measurability of a function [closed]
Is my proof for the following problem correct?
Let $f:X\rightarrow[0,\infty)$ and $X=\bigcup_{i\in\mathbb{N}}A_i$ be measurable.
Prove: $f$ is measurable $\iff$ $f\cdot\chi_{A_i}$ is measurable.
Proof:...
user1197316
-2
votes
1
answer
37
views
Integration of multivariate odd symmetric function [closed]
If $f(\mathbf{x}): R^p \rightarrow R $ is a multivariate odd-symmetric function in the sense that $f(\mathbf{x}) = -f(-\mathbf{x})$ for any $\mathbf{x}$ and it is absolutely integrable, does it ...
- 25
5
votes
1
answer
405
views
How to show that two integrals are equal?
In my advanced calculus course, I am struggling with the following problem where we need to show that the two integrals are equal.
Consider a function $g:[0,1] \times [0,1] \to \mathbb{R}$
defined by
$...
- 159
1
vote
1
answer
76
views
A 3N-variable integral with a condition
I have to calculate the following integral:
$$ \int \left( \prod_{i \neq j =1}^N |\vec{x_i}-\vec{x_j}| \right) \delta({\sum_{i=1}^N |\vec{x_i}|^2 - 1}) d^3x_1 d^3x_2 \cdots d^3x_N $$
where $N$ is a ...