# Questions tagged [multiple-integral]

For questions regarding computation and results related to integrals in at least 2 variables.

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### Define interval $M$ for inequality $(x^2+y^2)^2\leq2y^3$ and calculate integral $\iint_M1dxdy$

I first thought I could use polar substitution where $\phi$ would be $0\leq\phi\leq\pi$, but I couldn't figure out the bounds for the radius. I think it would be better to just define it normally ...
1 vote
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### Using multivector residue theorem to evaluate multiple integrals

I recently started learning about Geometric Algebra and Geometric Calculus. Since the residue theorem can be generalized for multivector functions I wondered if one could use it to evaluate certain ...
1 vote
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### Find volume of solid bounded by given surfaces. $z=a+x,z=-a-x,x^2+y^2=a^2$

Find volume of solid bounded by given surfaces. $$z=a+x, \qquad z=-a-x, \qquad x^2+y^2=a^2$$ This is the solid. We can find volume of solid that has positive $z$ value and multiply by $2$. And for ...
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### Solution to multidimensional gaussian integral with power factor

I would like to figure out for the derivation for a tensor ABCD optical law was derived for a flattened Gaussian beam in the following reference: https://www.sciencedirect.com/science/article/pii/...
1 vote
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### Triple integral set up using cylindrical coordinates

Set up an integral in cylindrical coordinates to evaluate $\iiint_{E} x y d V$ where $E$ is the region enclosed by the cone $z=2-\sqrt{x^{2}+y^{2}}$, the cylinder $x^{2}+y^{2}=1$, and the $x y$ plane. ...
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### What does $d^n\textbf{x}$ mean in this context?

I found the following on Wikipedia. Integration over more general domains is possible. The integral of a function $f$, with respect to volume, over an $n$-dimensional region $D$ of $\mathbb{R}^{n}$ ...
1 vote
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### Show example that if $\phi$ is not injective then the change of variables might yield different result

I need to find non-injective parameterization to some domain such that : $$\int_D f(x,y) dxdy \ne \int_E (f\circ \phi)(u,v)J(u,v) dudv$$ The example I found turns out to be wrong. (The integral is the ...
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### Double integral and change of variable [closed]

$$\iint_D \left(x^2-y^2\right)\ dxdy$$ over $D$ which is bounded by region enclosed by the four curves $y = x, y = x + 1, xy = 1$ and $xy = 2$ in the first quadrant. What will be a suitable change ...
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### Range of $\phi, \theta$ in $\int_0^{\pi/4} \int_0^{\pi/2} \int_0^{2\sin\phi \sin\theta} \rho^3\sin\phi \sin\theta d\rho d\theta d\phi$

The question: A solid bounded by the (y,z)-plane, the (x,y)-plane, the cone $x^2 + y^2 = z^2$, and the surface $x^2 + y^2 + z^2 - 2y = 0$. Suppose a density of a chunk of metal of the shape of this ...
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### Rewrite the integral in the order dy dz dx

Rewrite the integral $$\int_0^3 \int_0^{9-y^2} \int_\frac{y}3^1 f(x,y,z) dx dz dy$$ as an interated integral in the order dy dz dx. I have trouble visualizing if my answer is the correct iterated ...
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### Proofing the identity of double integrals over symmetric function

In one of my physics classes we had given a double integral over a function $f$ in two variables: $$G=\int_{x\in D}\int_{y\in D}f(x,y)\ \mathrm{d}y\ \mathrm{d}x$$ The function $f$ is symmetric in its ...
Given the sphere of radius two centered at the origin, you can slice a piece off by imposing the conditions that $z\ge 0$ and $y\ge 1$. I've been instructed to find the volume of this piece using ...