All Questions
Tagged with integration multivariable-calculus
5,463
questions
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Change the double integral from cartresian to polar cordinates
I have to solve this following integral,
$$\int_0^{\rm A} \int_{\sqrt{R_g^2 - x^2}}^{\sqrt{R^2 - x^2}+r_0} \arcsin{\left(\frac{R_{\rm g}}{\sqrt{x^2+y^2}}\right)} {\rm d}y {\rm d}x$$
Here ${\rm A}$, $...
- 141
2
votes
2
answers
63
views
Multiple Integral Problem with Dirac Delta Constraint: Seeking Guidance
I am working on a challenging multiple integral problem and would appreciate any assistance. The integral is as follows:
$$
\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \ldots \int_{-\infty}^{+\...
3
votes
1
answer
98
views
Area of a Quater-Circle with hyperbolic elements
The actual question states the following;
"Find the mass of a Quater-Disc (in terms of R), in the first quadrant, of radius 'R' if density varies as D = xy"
My first thought was somehow ...
- 109
5
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60
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Are there programs which compute integrals in $\mathbb{R}^n$?
I look for a software which computes integrals like, for example, this one:
$$\int_{B_R}\frac{|x|^3 dx}{(\varepsilon^2+|x|^2)^{n-1}},$$
where $\varepsilon>0$ and $R>0$ are not specific numbers ...
5
votes
1
answer
136
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General formula for reversing double integral bounds
The double integral over the region:
$$
R = \left\{ \left( x,\: y \right) : a \leqslant x \leqslant b,\: g\left( x \right) \leqslant y \leqslant h\left( x \right) \right\}
$$
is expressed as
$$
\...
- 170
0
votes
0
answers
111
views
Triple Integration of a Volleyball Serve
So I am a student that is working on a project that the end goal is to calculate the total volume of a 3D form created from quadratic equations that I already know and plotted in a Desmos 3D graphing ...
- 1
3
votes
1
answer
60
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Why is $\iiint_B(12x^2+2z)dxdydz=\iiint_B4(x^2+y^2+z^2)dxdydz$?
According to my teacher
$$\iiint_B (12x^2+2z) \, dx \, dy \, dz = \iiint_B 4(x^2+y^2+z^2) \, dx\, dy\, dz$$
where $B=\{(x,y,z)\mid x^2+y^2+z^2\le 1\}$. I have absolutely no clue why the triple ...
- 857
1
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0
answers
56
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Show equivalence of two very long integrals.
I'm trying to show that the following integral $$\int_0^\infty \int_{-\infty}^s f(x \vee s-b) \sqrt{\frac{2}{\pi}}\frac{1}{t^{\frac{3}{2}}}(2s-b)e^{-\frac{(2s-b)^2}{2t}-\mu(b+\frac{\mu t}{2})}db ds$$ ...
- 41
0
votes
5
answers
89
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Help with integration over a triangular region.
I'm currently trying to wrap my head around double integrals over a triangular region, when you are given the vertices of the triangle. I need to do the integral
$$
\iint_D 2e^{-y-x} \,\mathrm{d}y\,\...
- 21
1
vote
1
answer
61
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Integrate a sum of trig function under absolute value
Let $n \in \mathbb{N}$, I'm trying to compute an explicit formula for the following integral:
$$
\operatorname{I}\left(n\right) = \int_{\left[0,2\pi\right]^{\,\,n}}\,
\left\vert\rule{0pt}{4mm}\,{\cos\...
- 888
0
votes
1
answer
54
views
Why isn't the integral of the total differential of a function equal to the function? [duplicate]
For example let $w = 2x^3y$
then $$dw = \frac{\partial w}{\partial x}dx + \frac{\partial w}{\partial y}dy $$
$$\implies dw = 6x^2y\hspace{1.5mm} dx + 2x^3 \hspace{1.5mm} dy$$
Then why isn't it that $$...
- 11
0
votes
1
answer
49
views
Deal with discontinuity in double integrals
Let $f(x,y)= \frac{x^2+y^2}{x^2}$. Consider
$$ \int_D f(x,y)$$
with $D=\{ (x,y): 0\leq y\leq x, x^2+y^2\leq 1 \}$
The function is unbounded in $D$, due to the denominator. I checked that cannot be ...
- 499
1
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0
answers
35
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Solid bounded by five planes
Consider the solid limited by the three coordinates planes and the planes $x+y=1$ and $x+z = 1$. How can I set the limits so I can integrate in this region?
Ideas: the origin is include so $0\leq x, y,...
- 284
1
vote
1
answer
84
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Hint on examine the integrability?
How to examine the integrability of $f(x,y)=\left\{\!\!\!\begin{array}{c c}{{~x+y,}}&{{x,~y~\mathrm{are~rational},}}\\ {{~x-y,}}&{{\mathrm{other~cases.}}}\end{array}\!\!\right.$ on $[-1, 1]\...
- 123
2
votes
1
answer
65
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Can use one-variable integration to tell whether $R=\{(x,y):\;0\leq y\leq\left|\mathrm{sin}\frac{1}{x}|,\;0<x\ <1\right\}$ is rectifiable(has area)?
$R=\left\{(x,y):\;0\leq y\leq\left|\mathrm{sin}\frac{1}{x}\right|,\;0<x\ <1\right\}$, to determine whether it's rectifiable, can I use the method of one-variable integration to prove? Like this ...
- 123