All Questions
1,842
questions
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Simplification of Dependent Nested Integrals
I have been working with nested Integrals of the following form :
$$ \int_0^{T}dt_n \int_0^{t_n}dt_{n-1} \cdot \cdot \cdot \int_0^{t_2} dt_1 \prod_{i=1}^n f_i(t_i) $$
Some simpler cases and prework :
...
- 11
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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 ...
1
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0
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Introductory questions to fourier transform: help with the visualization of integration (with respect to one axis) of a 2D function
I'm not very familiar with more advanced concepts, but I have a decent understanding of single variable calculus with real-valued functions $f: \mathbb R \to \mathbb R$ (via Spivak). I wanted to learn ...
- 5,064
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54
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How to analyze the behavior of this function?
I have a map from positive reals to positive reals, of the form $$ f(d_1) = \frac{\int_0^\pi \int_0^\pi \int_0^{2\pi} 4\pi sin^2(\frac{\theta}{2})(1+(1-cos\theta )(v_1^2-1)) e^{k(d_1 + d_2 + d_3 + (1-...
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1
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62
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Evaluating $\displaystyle\iint\limits_{A}\left(x^3-y^2z^4,2y^3,z^3-3y^2z\right)\cdot\overrightarrow{n}~\mathrm{d}S$, where $A$ is the unit sphere
This is from UCHICAGO (GRE Math Subject Test Preparation), Week $5$, Problem $14$.
Let $A$ be the unit $2$-sphere in $\mathbb{R}^3$. Let $\overrightarrow{F}=\left(x^3-y^2z^4,2y^3,z^3-3y^2z\right)$ be ...
- 5,450
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29
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integration by parts $\int_{\mathbb{R}^N} -\Delta u (u-M)\varphi dx$
I'm trying to understand how it integrates by parts the following integral. This belong to the academic paper Stable solutions of $-\Delta u=f(u)$ in $\mathbb{R}^N$ with DOI 10.4171/JEMS/217. ...
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1
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1
answer
283
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Proving a lower bound for this double integral
Let $f:\mathbb R^n\to\mathbb R$ be a smooth function. Let $k>0$ and consider the cutoff function $T_k:\mathbb R\to\mathbb R$ defined for any $t\in\mathbb R$ as
$$T_k(t)= \int_0^t 1_{\mathbb R\...
- 3,277
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20
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How to integrate function involving gradient of multivariate function?
Let's say we have a function f(x,y) = $x^2 + y^2$ , where x and y are real numbers and I have to compute the integral
$ I = \int g(v) \triangledown_{v} f(v) dv $, where v = [x, y]$^T$ and g(v) is a $R^...
- 23
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1
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90
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How to solve the integral $\iint_D(x^2+y^2)^{-2} dxdy$ with where $D$ is $x^2+y^2\leq2, x\geq 1$ and after that the same integral but with $x\leq1$. [closed]
So my main question is after I use the Jacobian how do I write the new space $D^*$ when I substitute $x=r\cos θ$ and $y=r\sin θ$. I know that I will find what the range of $θ$ is by simply finding the ...
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37
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How to solve the $\iint_D \frac{(x-y)^2}{1+x+y}dxdy$ where D is the trapezoid with edges $(1,0),(0,1),(2,0),(0,2)$ and using $u=1+x+y$ and $v=x-y$
I know that you need to find the Jacobian of $u,v$ and multiply it with the existing $f(x,y)$ inside the integral. But how can I then find the limits of integration for $u$ and $v$ after that since ...
3
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1
answer
64
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Find volume of body between surfaces
Problem: Find volume of body defined as follows:
$z^2=xy$, $(\frac{x^2}{2}+\frac{y^2}{3})^4=\frac{xy}{\sqrt{6}}$, $x, y, z \ge 0$.
My solution:
So we're working in the all positive octant of the ...
- 41
4
votes
3
answers
134
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Find area of figure and volume of body
Problem
Find area:
$$\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)^2=\frac{xy}{c^2},\qquad a,b,c > 0$$
My solution:
The figure encloses area under itself so we're looking at: $$\left(\frac{x^2}{a^2}+...
- 65
2
votes
1
answer
45
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region above the cone and inside the sphere
I've come across a problem where
\begin{array}{c}
\mbox{I have to compute the volume of the region}
\\ \mbox{above the cone}\
z^2 = \sqrt{x^2 + y^2}\ \mbox{and inside the sphere}\ x^2 + y^2 + z^2 = 1
\...
- 75
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votes
2
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66
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Is $\int_D \nabla f dx = 0$ for $f$ compactly supported?
Let $D \subset \mathbb{R}^n$ be bounded and $f$ a smooth compactly supported function such that its support is contained within $D$. I am interested in
$$\int_D \nabla f dx.$$
If $n = 1$, then by the ...
- 6,277
3
votes
1
answer
115
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Computing an integral using differential under the integral sign
The following integral is in question.
$$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} $$...
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