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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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 ...
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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 ...
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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 ...
