# 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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### Integrating a vector field along the edge of a surface

At it for over a day now, must concede. Lets have a vector field $\vec{F}(x,y,z)=(xz,yz+x^2yz+y^3z+yz^5,2z^4)$ with the surface $\Sigma$ given as $(x^2+y^2+z^4)e^{y^2}=1, x\geq 0$ oriented so its ...
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### How to solve a surface integral

How should I find the boundaries for the following surface $S:$ $S$ is the surface with equations $z = x^2+y^2$ and $z\leq 1.$ I can find $x^2=1-y^2$ and take the square root to find the boundaries ...
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### How to evaluate$\int_{\partial D} \int_{\partial D} \frac{1}{|x-y|}dxdy$, where $D_1$ and $D_2$ are spheres in 3D?

In my previous question I asked about evaluating the following integral over the surface of a sphere in 3D: $$\int_{\partial D} \int_{\partial D} \frac{1}{|x-y|}dxdy,$$ and it turned out that the ...
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### Obtaining the surface area element geometrically

My text book derives a parametric formula for the surface area element $dS$ (which I follow and can understand). However it then goes on to remark that the surface area element can also be obtained ...
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### surface area using formula and double integrals

The prompt here is to find the surface area using double integrals. $$f(x) = 2\sqrt{xy}$$ with the vertices (1,1) (1,2) (2,2) (2,1). From resources the formula for surface area using double ...
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### Area of surface using polar coordinates

Calculate the area of the surface $z=x+y$ that is inside the cylinder $x^2+y^4 = 4$. I was able to find the correct answer by calculating the normal vector (using cross product) at each point on the ...
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### Finding a surface $S$ with property that the vector field $F$ has no work done on any path

$F(x,y,z) = (2x+y+z)i + (2y+z+x)j + (2z+x+y)k$. I previously proved that $F$ is conservative and so this means any closed loop on any piecewise differentiable surface evaluates the work done to be ...
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### Divergence theorem with nonorientable surfaces

The general theorem from which Gauss’s law stems, divergence theorem, states that the volume integral of the divergence $\nabla\cdot\bf F$ of $\bf F$ over $V$ and the surface integral of $\bf F$ ...
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### Finding surface parameterization of a cylinder?

I am new to multivariable calculus and am just get my head around the parameterization of surfaces. After research I found that a cylinder entered on the z - axis with radius R has a parameterisation ...
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### Closed curve divides into two or more regions compact and orientable surfaces with positive gaussian curvature

Proposition. Every closed curve divides into two or more regions every compact and orientable surface with positive gaussian curvature. I need to prove this sentences using Gauss-Bonnet theorem, but ...
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### Which paraboloid would have a larger surface area? Or would they be the same?

The enclosed area of these two paraboloids should be seen as a cross section of a 3D image. It is a water storage tank. I am trying to find out strengths and weaknesses of each model. Which one would ...
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### Evaluating a difficult surface integral

http://imgur.com/a/aHyz6 I need help solving this question, I know that you can get the parametrization of the ellipsoid but the curl of F seems so difficult to calculate, for part a. Am I missing ...
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### Computing the surface integral of a multivariate function

The problem reads: Find $\int_S \, f(x, z) \, dS$, where $f(x, z) = e^{-(x^2 + z^2)}$ and $S$ is the unit disk centered at the point $(0, 2, 0)$ and in the plane $y=2$. I'm not sure how to set ...
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### What is the surface area of part of the given plane?

The plane I am given is 3x+2y+z = 6, which lies in the first octant. I am having troubling figuring out my boundaries for the integral to plug into the surface area formula.
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### Flux through rotating cylinder using divergence theorem

A vector field $V$ has the divergence $div(V)=4$. What is the total flux in through the surface $\partial C$ of a massive rotating cylinder $C$ that has the height $h=2$ and the radius $r=5$? ...
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### Finding the flux by surface integrals and compute the volume

My attempt: (a) flux 1 = double integral (u cos v, u sin v, 1) . (0, 0, u) du dv = pi flux 2 = double integral ( u cos v, u sin v, u) . (-u cos v, -u sin v, u) du dv = 0 i think the answer for ...
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### Line integral for surface area of cylinder

I need help trying to use a line integral to find the lateral surface area of the part of a cylinder $x^2+y^2 =4$ below the place $x+2y+z =6$ and above the $xy$- plane I know how to find the ...
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### Gauss's divergence integration

I'm not sure on how to compute this integral using Gauss's divergence theorem. Can someone please explain. $\iint_S \vec{F} \cdot \vec{n}\quad dS$ $\vec{n}$ is outward normal and S is exterior ...
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### Area enclosed by a curve defined by parametric equations

A point set $B$ in the $(x,y)-$plane is determined by its parametric equation: $$r(u,v)=(u,10u^2+v\cdot u\cdot (5-10u)), 0\leq u\leq \frac{1}{2}, 0\leq v \leq 1$$ Determine the area of the ...
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### Surface Integral using divergence theorem over unit sphere?

Use the Divergence Theorem to compute the surface integral $\iint_T \vec{F}^{\,}* \,d\vec{S}^{\,}$ where T is the unit sphere $x^2+y^2+z^2=1$ and $\vec{F}^{\,}$ $<x,y,z> = <y, z, x>$. I ...
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### Stokes' Theorem to evaluate integral

Use Stokes' Theorem to evaluate $\int\limits_C (-y^3dx + x^2dy-z^3dz)$ where C is the interaction of cylinder $x^2+y^2=1$ and plane $x+y+z=1$. The surface on which the integral is to be done should ...
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### Is it possible to evaluate surface integral of flux across a closed surface when the integrand is constant

Suppose one wants to take the surface integral of outward flux $\mathbf{F(x)}$ where $\mathbf{x}=[x_1,x_2,x_3]$ across a spherical surface. $$\int\int_S \mathbf{F(x)}\cdot \mathbf{\hat{n}} \:\:dS$$ ...
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### Can normal vector be argument of the vector being surface integrated?

Given a scalar function (field?) $f(\mathbf{x})$ (where $\mathbf{x}=[x_1,x_2,x_3]$) with gradient $$\nabla f= [f_{x_1},f_{x_2},f_{x_3}]$$ Where $$f_{x_i}=\frac{\partial f}{\partial x_i}$$ And ...
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### Compute the flux of the vector field $F(x,y,z) = \left(2x-y^2\right) \mathbf i +\left( 2x - 2yz\right) \mathbf j + z^2 \mathbf k$
Compute the flux of the vector field $F(x,y,z) = \left(2x-y^2\right) \mathbf i +\left( 2x - 2yz\right) \mathbf j + z^2 \mathbf k$ through the surface consisting of the side and bottom of the cylinder ...