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.
31
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Bounding integrals using asymptotic expansions of the integrand
Im following the book "Analytic Combinatorics" by Flajolet and Sedgwick. I'm having trouble understanding the last part of the proof of the theorem regarding the standard function scale. In ...
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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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Help with change of variables for evaluating $\iint_S (x^2-y^2)e^{(x+y)^2} \,dx\,dy$
I have $$\iint_S (x^2-y^2)e^{(x+y)^2}\,dx\,dy $$ with restrictions $x+y\leq3$, $ xy\geq2$ and $y\leq x$
I think that with the variable changes $$u=x+y$$ and $$v=x-y$$
whose Jacobian is $2$
then I ...
- 31
2
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Estimate on integral
I'm working on my thesis, and in an article (link below) I came up with an estimate on an integral that I can't deal with.
Let $\gamma\in(0,1),$ and take some $k\in\mathbb{Z}^3,$ and let $|k|^2$ be ...
- 1,568
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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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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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Integral bounds for $x\geq yz$
I am having trouble understanding the integral bounds.
From what side should my understanding go (first or second?):
first: as $z$ is between zero and one, $y$ is also between zero and one, thus $x$ ...
- 2,527
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Stokes’ Theorem - On Conservative Vector Fields and Whether Parametrization is Always Necessary/Helpful?
In my physics class, we are working on problems using Stokes’ Theorem - but unfortunately we didn’t go over a single example Problem and it’s missing from our textbook! Now, when searching the ...
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Double integral limits
Evaluate the integral: $\iint_D xydA$ where D is union of the two regions
The integral is given as :
$\int\limits_{-1}^{1}\int\limits_{0}^{\sqrt{1-x^2}}xydydx+\int\limits_{-\sqrt3}^0 \int\limits^{\...
- 282
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triple integration to find volume
how can I solve this problem I already have the final answer but I don't know how to solve it , the answer for this problem is $(V=8(pi-1/3)$
I tried solving it with this:
$\int_{-2}^2\int_{-\sqrt{4-...
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Fixing the bounds of an indefinite integral
Question:
$$\frac{\partial^2 u}{\partial x^2}-\frac{\partial^2 u}{\partial t^2}=f(x,t) \qquad \qquad u(x,0)=\frac{\partial u}{\partial t}(x,0)=0$$
Solve this equation, writing the solution in the ...
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Remainder in Asymptotic Expansion of Erfc
According to Abramowitz and Stegun: Handbook of Mathematical Functions (7.1.23 and 7.1.24, http://people.math.sfu.ca/~cbm/aands/page_298.htm),
we have Asymptotic expansion of Erfc is given by
\begin{...
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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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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 ; ...
- 143
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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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