Questions tagged [bounds-of-integration]
In many questions the problem of determining bounds of integration in multiple integrals is a major part of what an answer needs to deal with, and in surprisingly many questions it is the only issue. This tag is for such occasions.
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How to get a CDF value from a PDF when the required CDF is not within the defined area?
I have a density function f(x, y) = 1/2 for 0 ≤ x ≤ y ≤ 2 and 0 elsewhere. I am being asked to find the CDF value F(1, 3), but as you can see the three is past the range of the defined triangle, what ...
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triple integral pyramid bounds
I am still confused about how to set up bounds for double and triple integrals. My task is to set up bounds for a function that is a pyramid with edge coordinates $(5,+-5, 0)$, $(-5,+-5,0)$, $(0,0,4)$....
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How does one calculate the area of a set?
The set is $M=\{(x,y)\in\mathbb{R}^2:|x|+|y|\leq 1\}$.
Question: How do you calculate the area of $M$? More specific, how do you find the bounds of integration?
Attempt: I tried to solve the ...
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Changing the Order of Integration in a Triple Integral
I'm currently studying for my multivariable calculus exam and I've come across a problem that I can't seem to solve. I have a triple integral with the order of integration $dz \, dy \, dx$ and I need ...
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An upper bound for an integral
I saw many references using the following estimate but I couldn't prove it.
Given $T>0$ and $0 < b \leq \frac{1}{2}$, exist $C(b)$ constant that depends only on $b$ such that
\begin{equation}
\...
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Why don't the bounds in this definite integral change?
The question
This is probably a very basic question but I'm having a brain lapse and don't know why they didn't change the definite integral bounds from ($0 \rightarrow4$) to ($4 \rightarrow20$). I ...
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Question on the bounds of definite integration during a substitution
Apologies if this question is rather elementary. I seem to still misunderstand a few things about how bounds change during substitutions still.
I was taught in calc II that to perform a substitution, ...
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How do I find the bounds of this particular integral?
I want to convert this integral to Polar Coordinates to solve it: $\int_{0}^{2}\int_{0}^{\sqrt{y}}4xy^{2} \, dx \, dy$
What would be the bounds of $r$ and $\theta$ be?
I know how to solve the integral ...
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Evaluation of $~\iint_{D} {x^2-y^2 \over 1+x^4+y^4 } \mathrm{d}x \mathrm{d}y~$ where $~D~$is of bounded and closed and line symmetric with $~y=x$
$~D:=$domain where it is bounded and closed in $~ \mathbb R^2 ~$ and line symmetric with $~ y=x ~$
$$
I:=\underbrace{\iint_{D} {x^2-y^2 \over 1+x^4+y^4 } \mathrm{d}x \mathrm{d}y}_{\text{I want to ...
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Bounding an exponential row
Let $0<c<1$. I need to bound
$$
\sum_{i=1}^n \frac{c^{n-i}}{i}\leq C n^{-?}
$$
for some constant $C>0$. Does anyone know how to optimal bound this sum?
Thank you very much for any suggestions....
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Prove that $\int_0^1 e^{-tu}(1-u)^{\alpha}du\leq t^{-1}$ for $\alpha,t>0$
Let $\alpha>0$, I need to prove that there exists $t_0>0$ such that $$\int_0^1 e^{-tu}(1-u)^{\alpha}du\leq t^{-1}, \forall t>t_0.$$ I received help and found that by Watson's Lemma you could ...
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Show that the sum of two integrals is finite.
How to easy show that
\begin{equation} \int_0^1 \frac{1-e^{-x}}{x}dx+\int_1^M\frac{-e^{-x}}{x}dx \end{equation} is less than finite number?
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Calculus: Finding Volume with Triple Integrals
**Problem:**A shape is bounded by the following elliptical function $4x^2 + y^2 +z = 128$ and the planes $x=0, x=4, y=0, y=4$. Find the volume of the shape.
My attempt:
$4x^2 + y^2 +z = 128 \implies z ...
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Double integral of $1/(x^2+y^2)$ restricted to $x^2+y^2\leq2$ and $x\leq1$
Find $$\iint_D \frac{1}{(x^2+y^2)^2}dA$$ where $$D = \left\{ (x,y): x^2 + y^2 \leq 2 \right\} \cap \left\{ (x,y): x \geq 1 \right\}$$
Because of the prevalence of $x^2+y^2$ terms here, I figured we ...
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HLS inequality not suitable to bound this integral
I am trying to bound the following integral for $f \in L^{n/2}(\mathbb{R}^n)$:
$\int_{\mathbb{R}^n}\int_{\mathbb{R}^n} f(x) \lvert x-y \lvert^{2(2-n)}f(y)dxdy$. Because of the factor 2 in the exponent,...
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