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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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 $x$ ...
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Why does changing integral bounds get me the wrong answer?
Full disclaimer, this is a homework question.
While solving this question, I came upon the integral $$\int_{-r}^{r}\frac{b\tan^{-1}(\theta)}{2}\sqrt{r^2-x^2} dx$$ Proceeding with trig substitution I ...
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Help me understand easy (not for me) concepts in volume integral
Keep looking at the page for an hour.
Still not sure how I can get the sloping surface of $x+y+z=1$ and integration ranges for $x, y, z$. and why their range is different too.
The book keeps throwing ...
user808793
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Evaluate $\int \int \int_B x^2+y^2 \, dxdydz$
I'd like to evaluate $\int \int \int_B x^2+y^2 \, dxdydz$ where $B$ is the area enclosed by $x^2+y^2=2z$ and $z=2$ but I'm not sure about the bounds. I've thought something like this...
$$\int_0^2 \...
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Determine whether the improper integral $\int_{0}^{\infty}\frac{x^3}{1+x^4}\,dx$ exists
While doing an exercise I need to prove that $\frac{x^3}{1+x^4}$ is integrable.
So I have to see if $\int_{0}^{\infty} |\frac{x^3}{1+x^4}| dx < \infty$. I tried to divide it in two integrals but I ...
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Volume bounded between sphere and three planes
I found a question in my homework that I have been trying to solve for days with minimal progress. We're given a sphere of form $x^2+y^2+z^2=9$ and three planes, $x=1,y=1,z=1$
The sphere in question:
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Double integral setup - Uniform distribution
I am currently trying to understand a specific component of a probability problem involving setting up the proper bounds on a double integral.
In the problem, $X_1$ and $X_2$ are independent Uniform $(...
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Bounds for marginal density function
I have the joint density function $f(y_1,y_2) = 3y_1, 0 \leq y_2 \leq y_1 \leq 1.$ And $0$ elsewhere.
I have to find the marginal density function for $y_2$
My question is how to define the right ...
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Upper-bound on summation of form $\sum_{i=1}^{t} \frac{\alpha^{t-i}}{\sqrt{i}}$, where $\alpha <1$.
Suppose we are given a fraction $\alpha <1$. My question is whether we can derive an upper bound on summations of the following form:
$$S_t= \sum_{i=1}^{t} \frac{\alpha^{t-i}}{\sqrt{i}},$$ where $\...
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$\iiint_M (x+y+z)\,dx\,dy\,dz$ over $M=\{(x,y,z)\in\mathbb{R^3}: 0≤z≤(x^2+y^2)^2≤81\}$
$$\iiint_M (x+y+z)\,dx\,dy\,dz$$ over $M=\{(x,y,z)\in\mathbb{R^3}: 0≤z≤(x^2+y^2)^2≤81\}$. How would I express this with the correct bounds? Once I have the bounds I can continue on my own but I need ...
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How do you find the bounds for a joint probability distribution function?
$$\begin{aligned} f(x, y) &=\begin{cases}1/(x^{2} y^{2}) & \text { für } &x \geq 1, y \geq 1 \\[1ex] 0 &&\text { sonst. }\end{cases}\\[2ex] V&:=X Y\end{aligned}$$
Find the ...
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Double Integral: how to write the bounds for this triangular region?
Let $0<a<b$ and consider the triangular region bounded by the three points $(a,a)$, $(b,a)$ and $(a,b)$.
If we would integrate some function $F(x,y)$ over this region, how does one write down ...
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How can I find $\iiint\frac{xz}{1+x^2+y^2}\,dz\,dy\,dx$ where $1≤x^2+y^2≤3, 0≤z≤3$?
Compute $$\iiint\frac{xz}{1+x^2+y^2}\,dz\,dy\,dx,$$ where $1≤x^2+y^2≤3, 0≤z≤3$.
I've tried it. But I'm only confused with $\theta$. I think it should be $0$ to $2\pi$, but that'll make the whole ...
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How can you simplify/verify this solution for $\int\limits_0^{.25991…} Q^{-1}(x,x,x)dx?$
As I do not know the complex behavior of this function, it would be even harder to integrate past the real domain. The upper bound for the domain is a constant I will denote β.
$${{Q_2}=\int_0^βQ^{-1}(...
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Quick question about integration of a piecewise defined function
I have the following "quick" question about the piecewise function integration:
Say, I have to find $\int\limits_0^{1}f(x)d x$, with $f(x)$ being piecewisely defined on $\mathbb{R}$ as ...
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