# Questions tagged [surface-integrals]

In mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral.

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### Calculating a surface integral of a surface by another surface

Let $\Sigma$ be the surface $\frac{x^2}{2}+y^2=1$, $0\leq z\leq 1$ and $u(x,y,z)=\frac{x}{x^2+y^2},\frac{y}{x^2+y^2},z^2-z)$. Calculate the flux integral $\iint_\Sigma (u|N)dS$ where $N$ points away ...
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### Area of the part of the cylinder $x^2+y^2=2ay$ outside the cone $z^2=x^2+y^2$

Problem: Find the area of the part of the cylinder $x^2+y^2=2ay$ that lies outside the cone $z^2=x^2+y^2$. My attempt: So I thought we could do this by projecting the surface onto the $yz$-plane and ...
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### How to prove $\oint_{A}\mathbf{E}\cdot d\mathbf{A}=\frac{q}{\epsilon_{0}}$ mathematically and rigorously?

How to prove $\oint_{A}\mathbf{E}\cdot d\mathbf{A}=\frac{q}{\epsilon_{0}}$ mathematically and rigorously? This equation is called “Gauss's Law” in physics. I seek for a rigorous mathematical proof for ...
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### Calculating Electric Flux Through a Closed Surface

I'm trying to solve a problem involving the calculation of electric flux through a closed surface, but it's my first time attempting such a problem and I could use some guidance. Any help would be ...
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### The surface integral $\iint_S (z^2 + y^2 + x^2) \, dS$ over the cube $S$? [closed]

Evaluate the integral $$\iint_S (z^2 + y^2 + x^2) \, dS ,$$ where $S$ is the surface of the cube $\{-a < x < a, -a < y< a, -a< z< a\}$. I've attempted to partition the surface into ...
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### Surface integral over the surface of the cone $z=1-\sqrt{x^2+y^2}$ lying above the $xy$-plane and normal making an acute angle with $\vec k$

Let $\vec F=(x^2+y-4,3xy,2xz+z^2)$ and $S$ be the surface of the cone $z=1-\sqrt{x^2+y^2}$ lying above the $xy$-plane and $\vec n$ is the unit normal to $S$ making an acute angle with $\vec k$ , then ...
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### Finding the volume of 3 dimensional region under the graph of a function.

Im trying to do the following question but im confused. Let W be the three dimensional region under the graph of the function $f(x,y) = \mathrm{e}^{x^2+y^2}$ and over the region in the $(x,y)$ plane ...
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### Surface Integral Over Part of Plane

Let $S$ be the surface where $x + 2y + 3z = 0$ and $-1 \leq y \leq 1, 0 \leq z \leq 1$. Compute the surface integral $$\int \int_S (2x,3y-x,1-2y) \cdot \mathbf{\hat{N}}dS$$ where the unit normal ...
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### Computing the surface integral of $\ G(x,y,z) = xyz$ over the triangular surface with vertices $(1,0,0),\,(0,2,0)$ and $(0,1,1)$.

I was attempting to compute the surface integral of $\ G(x,y,z) = \ xyz$ over the triangular surface with vertices at $\ (1,0,0)$, $\ (0,2,0)$ and $\ (0,1,1)$. Clearly, the first step is to ...
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### Evaluate Surface Integral over this triangular surface

When I solving the practice exercise problems at the end of the section, I stumbled upon this problem, which I have been trying to figure out how to compute the integral, but couldn't. Can someone ...
Evaluate the integral $\iint xyz\,ds$ where $S$ is the triangle with vertices $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$. Well, I did almost everything, I'm just stuck at finding the boundaries. Firstly, I ...