# 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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### Unable to evaluate this surface integral

Question is to find the surface integral $$\iint p(x^4 + y^4 + z^4) \, \mathbb dS.$$ The given surface is $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1,$$ and $p$ is the length of the ...
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### Evaluating the surface integral over subset of Cylinder surface

I am preparing for the GRE math subject test. And I am not sure whether my solution to this surface integral problem is correct. Problem Image. Surface $S$ is part of $x^2 +y^2 = 1$ between ...
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### Computing the average of $\prod_i (1-\frac{|x_i|}{L})$ on the surface of the unit $n$-sphere

I'm trying to compute the expected value of $\prod_i (1-\frac{|x_i|}{L})$ on the surface of a $n$-dimensional sphere. A first step could be to integrate only on the first quadrant to take out the ...
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### Apparent (minor) error in Cauchy's article on pressure or tension in a solid body

In his article De la pression ou tension dans un corps solide [On the pressure or tension in a solid body], Cauchy introduces a theory that allows to define Cauchy stress tensor. It looks like he ...
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### Surface area of a sphere above the plane $z=1$

The question is: Find the surface area of the part of the sphere $x^2+y^2+z^2=4$ that lies above the plane $z=1$. I got $4\pi(\sqrt3-1)$ but the answer key says $4\pi(\sqrt2-1)$. Am I doing something ...
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### How to evaluate this integral over a slice of the unit disk?

I have a function $f(r)$ in polar coordinates, for some positive $a$ and $b$, with $0<n<1$, defined on the unit disk ($0<r<1$) $$f(r) = a + \dfrac{b}{(1-r)^n}$$ I would like to integrate ...
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### How to Calculate the flux of the Vector Field on the surface $z = 1-x^2-y^2$ ( getting normal vector $(0,0,0)$ at the point $(0,0,1)$ ?!!! )

Let $S$ be the surface $z = 1-x^2-y^2 , 0\leq z$. Find $\int_{S} x^2z~dydz + y^2z~dzdx + (x^2+y^2)~dxdy$. Choose the direction of the normal upwards. so i calculated the flux and i got that it ...
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### How to find the surface integral of Torus intersecting with cylinder ??

Let $T$ be the torus obtained by revolving the circle {$(x,0,z)| (x-3)^2 + z^2 = 1$} about the $z$-axis. Find the area of the surface obtained by taking the intersection of $T$ with the ...
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