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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Prove/disprove upper bound and lower bound of the Integral
Hey I need to Prove or disprove this sentence:
$$
\frac{4}{9}(e-1) \leq \int_0^1 \frac{e^x}{(1+x)(2-x)} \, dx \leq \frac{1}{2}(e-1)
$$
using the infimum and supremum method for integrals, where m and ...
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Calculating Volume of Spherical Cap using triple integral in cylindrical coordinates and spherical coordinates
Given a sphere above of $xy$-plane with center $(0,0,0)$ and radius $2$ (the equation $z=\sqrt{4-x^2-y^2}$). Plane $z=\sqrt{2}$ intersect the sphere.
I want calculate volume of spherical cap (orange ...
- 2,354
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Write triple integral as cylindrical coordinate of given region, confused in determining lower and upper bound.
Given $E$ is a region as follows:
$$E=\left\{0\leq x\leq 1, 0\leq y\leq \sqrt{1-x^2}, \sqrt{x^2+y^2}\leq z\leq \sqrt{2-x^2-y^2}\right\}.$$
Write triple integral
$$\iiint_\limits{E}xydzdydx$$
as triple ...
- 2,354
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how to write this region $D$ in relation to $r,\theta$ in this $\iint_Df(x,y)dxdy$ where $D=\{x^2+y^2 \le1,x+y\le 1\}$ and $D=\{x^2+y^2\le1,x+y\ge1\}$
I have attached two photos showing the integration bounds and I find it tricky how to express $r$ and $\theta$ in those two, if $x=r \cos{\theta}$ and $y=r\sin{\theta}$, so any help is very much ...
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Probability densities with conditions - how to find the distribution function
I have two probability density functions where i need to find the distribution function.
The first function is
$$f(x,y)=
\begin{cases}
\frac{x}{y} & \text{for $0\leq x\leq y\leq c$}\\
0&\text{...
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Triple Integral - Use symmetry for center of mass question?
I am unsure when to use symmetry with triple integrals.
Can I use symmetry for this centre of mass question?
$E$ is bounded by the parabolic cylinder $z=1-y^2$ and the planes $x+6z=6, x=0$, and $z=0 ; ...
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Triple integral (mass) - setting up region between planes and parabolic cylinder
I am trying to set up the following triple integral using the xy plane.
$E$ is bounded by the parabolic cylinder $z=1-y^2$ and the planes $x+6z=6, x=0$, and $z=0 ; \quad \rho(x, y, z)=8$.
I set up ...
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How can I evaluate the bounds of this integral?
I have got this integral from a fourier transform:
$$\int_{-\infty}^0 e^{-ikx+x/4}+\int_0^{\infty} e^{-ikx-x/4}dx$$ Apparently the integrals give:
$$=\frac{1}{1/4 -ik}+\frac{1}{1/4 +ik}$$
But how? I'm ...
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Uniform initial conditions make Fokker-Planck/Kolmogorov Equation boundary conditions inconsistent
When considering the time evolution of a distribution over a state variable $x$, one of the cases that seems fundamental is when knowledge about $x$ begins uniform. However, modelling the process as ...
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Double integral: technique to derive the limitations of $y$ (or $x$)
$$\iint_Cy dxdy, \quad C=\{(x,y)\colon0\leqslant x\leqslant2+y-y^2\}.$$
It is simple to see $x=2+y-y^2$ is a parabola with the symmetry axes is $x$ and the vertex $(9/4,1/2)$. It is easy to find the ...
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How is this property of definite integral derived?
The property:
$$
\int_a^b f(x) \, dx=\int_a^b f(a+b-x) \, dx
$$
Derivation given in my textbook:
Let $t = a+b-x$. Then $dt = -d x$. When $x=a, t=b$ and when $x=b, t=a$. Therefore,
$$
\begin{aligned}
\...
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Upper bound of the integral $\int_\delta^\infty t^m e^{-\nu t^2} dt$
I am reading Wong's book on "Asymptotic Approximations of Integrals". On page 497, the book recalls (without proof) the following estimate: for all $\delta>0$ and $\nu>1$,
$$
\int_\...
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For which lower bound of integration $a$ does a definite integral of $f(x)$ from $a$ to $x$ equal its antiderivative $F(x)$ with $C=0$?
For an arbitrary antidifferentiable function $f(x)$, my goal is to construct a definite integral of $f(x)$:
$$
\int_a^x f(t) dt
$$
which is equal to one of the infinitely-many antiderivatives of $f(x)$...
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Find the density of random variable $X+Y$ for $f(x,y) = 6(x-y)$ if $0 \leq y \leq x \leq 1$
Find the density of random variable $X+Y$ for $f(x,y) = 6(x-y)$ if $0 \leq y \leq x \leq 1$.
I am able to use method of transformation to convert $f(x,y) = 6(x-y)$ if $0 \leq y \leq x \leq 1$, and ...
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Control $\int_0^\infty |\psi(x)|^2 dx$ by $\int_0^\infty \int_0^\infty K(x+y)\psi(x) \psi(y) dxdy$
Assume that $\psi(x)$ is bounded and integrable on $x \in [0,\infty)$ with $\int_0^\infty \psi(x) dx = 0$, and suppose that $K \colon (0,\infty) \to (0,\infty)$ is some kernel function satisfying $K(x)...
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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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2
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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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1
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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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