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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Double Integral $\int\limits_0^1\!\!\int\limits_0^1\frac{(xy)^s}{\sqrt{-\log(xy)}}\,dx\,dy$
Is it possible to get a closed form of the following integral?
$$I=\int_0^1\!\!\!\int_0^1\frac{(xy)^s}{\sqrt{-\log(xy)}}\,dx\,dy\quad\quad\quad(s>0).$$
My attempt: I’ve tried a change of variables ...
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When is $\int_a^b \frac{1}{x}\ln\bigg(\frac{x^3+1}{x^2+1}\bigg)dx=0$?
I would like to find positive, distinct, algebraic real numbers $a,b\in \mathbb R^+\cap\mathbb A$ satisfying
$$\int_a^b \frac{1}{x}\ln\bigg(\frac{x^3+1}{x^2+1}\bigg)dx=0$$
Does anyone know of a ...
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Is it necessary to write limits for a substituted integral?
To solve the following integral, one can use u-substitution:
$$\int_2^3 \frac{9}{\sqrt[4]{x-2}} \,dx,$$
With $u = \sqrt[4]{x-2}$, our bounds become 0 and 1 respectively. Thus, we end up with the ...
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Change of variables Double integral
I have
$$\iint_A \frac{1}{(x^2+y^2)^2}\,dx\,dy.$$
$A$ is bounded by the conditions $x^2 + y^2 \leq 1$ and $x+y \geq 1$.
I initially thought to make the switch the polar coordinates, but the line $x+...
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When does integration via u-substitution break down, equal limits of integration?
Edited: changed $\displaystyle\int_{a}^{b}f(g(t))g'(t) \, dt = \int_{g(a)}^{g(b)}f(x) \, dx$ TO $\displaystyle\int_{a}^{b}f(t) \, dt = \int_{f(a)}^{f(b)}u \, \frac{du}{f'(f^{-1}(u))}$
I am wondering ...
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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.
$${{Q_2}=\int_0^βQ^{-1}(...
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Is it correct to write $\int_a^x f(x) dx$?
The question pretty much sums it all. A few days ago when studying how to find the real part of a function knowing the imaginary part (or vice versa) I was given this formula: $$u(x, y) =\int_{x_0}^{...
user556151
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Bounds for the Harmonic k-th partial sum.
I need to bound the k-th partial sum or the Harmonic series. i.e.
$$ln(k+1)<\sum_{m=1}^{k}\frac{1}{m}<1+ln(k)$$
I'm triying to integrate in $[m,m+1]$ in the relation $\frac{1}{m+1}<\frac{1}...
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How to calculate the floor integral $\int_0^{\pi}\lfloor\pi^2\cos^3x\rfloor\sin x\,dx$?
$$\int_0^{\pi}\lfloor\pi^2\cos^3x\rfloor\sin x\,dx$$ (where $\lfloor x \rfloor $ is the floor of $x$)
I thought of breaking into required bounds but its too lengthy. Moreover I had to take cube root ...
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Double Integral $\iint\limits_D\frac{dx\,dy}{(x^2+y^2)^2}$ where $D=\{(x,y): x^2+y^2\le1,\space x+y\ge1\}$
Let $D=\{(x,y)\in \Bbb R^2 : x^2+y^2\le1,\space x+y\ge1\}$. The integral to be calculated over $D$ is the following:
\begin{equation}
\iint_D \frac{dx\,dy}{(x^2+y^2)^2}
\end{equation}
I do not know ...
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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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$\iiint_M (x+y+z)\,dx\,dy\,dz$ 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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Evaluating $\int_{-4} ^4\int _0 ^{\sqrt{16-x^2}} \int _0 ^{16-x^2-y^2} \sqrt{x^2 + y^2}\,dz\,dy\,dx$
Question: Evaluate the given triple integral with cylindrical coordinates:
$$\int_{-4} ^4\int _0 ^{\sqrt{16-x^2}} \int _0 ^{16-x^2-y^2} \sqrt{x^2 + y^2}\,dz\,dy\,dx$$
My solution (attempt): Upon ...
user776254
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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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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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