All Questions
Tagged with integration multivariable-calculus
5,463
questions
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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.
$$\int_0^1\int_0^1\frac{\partial^2}{\partial\rho^2}\{s\rho^2C(\textbf{r}_2,ss^\prime\rho)...
- 3,234
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Simplification of Dependent Nested Integrals
I have been working with nested Integrals of the following form :
$$ \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) $$
Some simpler cases and prework :
...
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Looking for citation of a definition of multivariable integral
I need a definition of Multivariable Riemann Integral to cite in my article.
I've been searching different cited sources, for example Rudin, Folland, Stewart and Larson books.
In the first two I ...
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Am I correctly applying repeated integration by parts?
Say we have two compactly supported functions $f,g:\mathbb{R}^n\to\mathbb{R}$. I found myself computing
\begin{equation}
\begin{split}
\int_{\mathbb{R}^n}f\frac{\partial^{r}g}{\partial x^{\alpha_1}\...
- 5,166
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Need help with the steps and limits in this multivariable integration of a joint probability density function
I am not sure how to proceed with this double integration. I know this can be evaluated in a much easier way than solving the integral as its just the volume of a cube but I need help with the process ...
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Intergrating $ \int_{M} f (x.,y,z,w)\ d {\rm Vol}_3 $
I want to integrate $$ I = \int_{M} f (x.,y,z,w)\ d {\rm
Vol}_3 $$where $f(x,y,z,w) = (x+y)e^{z+w} $, and
$M = \{x+y+z+w = 1, x,y,z,w > 0\} $.
I need to find a parameterization of M; if I consider $...
- 118k
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answer
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Integrate over sinusoidal region $\int\int(1 - \cos 2\theta ) (1 + \cos(\frac{\theta + 2 \pi u}2 )) du \,d\theta$
I'm trying to obtain a closed form expression for the following definite integral in $\mathbb{R}^2$:
$$
\int_{0}^{\pi} \int_{u = 0}^{u= A \sin\theta} f(\theta, u)\,\mathrm{d}u \,\mathrm{d}\theta ~,\...
- 268k
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How to find the following limit? $\lim\limits_{t \to {\pi}/{2}}\frac{ \int_{\sin t}^{1}e^{x^2\sin t}dx}{\int_{\cos t}^{0}e^{x^2 \cos t}dx}.$
I am trying to solve the limit $$\lim\limits_{t \to \pi/2}\frac{ \int_{\sin t}^{1}e^{x^2\sin t}dx}{\int_{\cos t}^{0}e^{x^2 \cos t}dx}$$
My first method was to try with L'Hopital, i derived using ...
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Question about Lemma 19.1 in Munkres' Analysis on Manifolds
In Munkres' Analysis on Manifolds, page 162 Lemma 19.1 Step 2 it states: Third, we check the local finiteness condition. Let $\mathbf{x}$ be a point of $A$. The point $\mathbf{y}=g(\mathbf{x})$ has a ...
- 18.6k
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Integration by parts on an area
I'm reading an Engineering book. All I can think of is integration by parts
$$\int_{\Omega} \dfrac{\partial M_y}{\partial x}v\text{d}\Omega = M_y v|_{?}^{?} - \int_{\Omega} M_y\dfrac{\partial v}{\...
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Introductory questions to fourier transform: help with the visualization of integration (with respect to one axis) of a 2D function
I'm not very familiar with more advanced concepts, but I have a decent understanding of single variable calculus with real-valued functions $f: \mathbb R \to \mathbb R$ (via Spivak). I wanted to learn ...
- 5,064
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2
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4k
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Volume bounded by sphere and plane
Find the volume of the region B bounded above by the sphere $x^2 + y^2
+ z^2 = a^2$ and below by the plane $z = b$, where $a > b > 0$.
Here I worked out $z=\sqrt{a^2-r^2}$ and $x^2+y^2=a^2-b^2$...
CommunityBot
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1
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Proving a lower bound for this double integral
Let $f:\mathbb R^n\to\mathbb R$ be a smooth function. Let $k>0$ and consider the cutoff function $T_k:\mathbb R\to\mathbb R$ defined for any $t\in\mathbb R$ as
$$T_k(t)= \int_0^t 1_{\mathbb R\...
- 3,277
5
votes
1
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Integral calculated directly and with Gauss Green formula
I checked on Wolfram Alpha that the directly calculated integral was correct, when I go to use the Gauss Green formula, I get $0$ as a result, when instead, I ...
- 5,840
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2
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Theorem 16.5, Munkres' Analysis on Manifolds [closed]
In Munkres' Analysis on Manifolds, page 142 Theorem 16.5 it states:
$$\int_{D}f\leq\int_{A}f$$
at the end of that page.
Here, $D=S_{1}\cup\cdots\cup S_{N}$ is compact since $S_{i}=Support\phi_{i}$ and ...
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