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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.

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. Given a surface, one may integrate over its scalar fields (that is, functions which return scalars as values), and vector fields (that is, functions which return vectors as values).

Read more on wikipedia's entry Surface integral.

1384 questions
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Generalizing a surface integral to 4 dimensions

I am trying to evaluate a surface integral, but instead of using a surface in $\mathbb{R}^3$, using a surface in $\mathbb{R}^4$. That is to say, $\oint_S f(x,y,z,w)\,dS$, where S is given by some $r(u,v,t) = \left( x(u,v,t) , y(u,v,t) , z(u,v,t) ,…
user817934
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Flux through the Surface

Find the flux through the surface $\iint_S F\cdot NdS$ where N is the normal vector to S. i) $F=3z\hat i-4\hat j+y\hat k$ $~~~S:z=1-x-y$ (first octant)   ii) $F=x\hat i+y\hat j-2z\hat k$ $~~~~S:\sqrt{a^2-x^2-y^2}$ I have evaluated $N$ vector as :…
Aladdin
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Show that $\iint_S (x^2+y^2) dA = 9 \pi /4$

In exam it was asked to show that $$\iint_S x^2+y^2 dA = 9 \pi /4$$ for $$S = {\{(x, y, z) | x>0, y>0,3>z>0, z^2 = 3(x^2 + y^2)}\}$$ I have tried many times but I don't get the $9 \pi /4$. $$\begin{align} \iint_S\sqrt{1+f_x^2+f_y^2}\,dA…
Emma
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Stokes' theorem on unit disc

I'm stuck with this question: "Consider $\iint _S(\vec \nabla \times \vec F)\cdot d\vec S$, where $\vec F=xyz\vec i+y^2z\vec j+xy^2\vec k$ and $S$ is the surface of the unit disc $0\le x^2+z^2\le 1$ on the plane $y=1$, with the normal to the plane…
user248052
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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 wrong or is the answer key wrong? Thanks in…
Gabriel
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Find the flux of the curl across a surface

I'm solving the below exercise. Find the flux of the curl across $S$ in the direction $n$ of the field $F$, when $S : z = x^2 + 4y^2$ lying beneath the plane $z=1$, with the normal having a positive $k$-component, and $F = \langle y, -xz,…
이심심
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To evaluate a surface integral

I was stuck in a question as follows: $$\int_0^R\int_0^R\frac{(R^2-x^2)(R^2-y^2)}{(x+y)^2}dxdy$$ I tried a lot to simply this but I was unable to do so. I am encountering surface integrals for the first time. I can't make use of the symmetry in $x$…
shsh23
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calculate surface integral of a cylinder with rectangular vector

I am lost with that problem, and I cannot continue. The problem asks to calculate that: $$\oint_S\ \vec F\cdot \vec {dS}$$ Being a vector defined by: $$F = x^2\hat a_x+ y^2\hat a_y + (z^2-1)\hat a_z$$ and S is defined by these Cylindrical…
user2793412
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How to interpret negative surface area?

In calculating $\iint_Dx^2y-y^5 dxdy$ where D is given by:$$~~~~~1-y^2\leq x\leq 2-y^2\\-\sqrt{1+x}\leq y\leq\sqrt{1+x}$$ I refered to the graphs in the following link: Desmos_1 to determine the region of integration and reverse the order of…
TheMercury79
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Is this a correct way to solve this surface integral?

So I was told to find the surface integral $$\iint_{S} yz \mathrm dS$$ for the surface $S$ parametized as $$x=u^2,y=u \sin{v}, z=u \cos{v}$$ for the region $0\leq u\leq1$ and $0\leq v \leq \pi/2$. So whati did was I computed the magnitude of the…
tsp216
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Notation for surface integrals over vector fields

Why do every question I see for a surface integral over a vector field $F$ under the surface $S$ be denoted by $$\int_{S} F \cdot n dS$$? As in, why does it only have a single integral? Where as once we expand the $dS$ to be $dudv$ (our parametric…
Morris C.
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Geometrically, why is this surface integral $0$?

Question is in Paul's Math Online Notes Q3: http://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceIntegrals.aspx It says: Evaluate $\int \int_S y \: dS$ where $S$ is the portion of the cylinder $x^2 + y^2 = 3$ between $z=0$ and $z=6$. I know that…
Twenty-six colours
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Is this problem ill-defined?

I am confused by a problem from Schey's Div, Grad, Curl and All That: Use the divergence theorem to show that $$\iint_S\hat{\mathbf{n}}dS=0,$$ where S is a closed surface and $\hat{\mathbf{n}}$ the unit vector normal to the surface $S$. The…
Lachy
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Surface integral (Please help me)

Consider a region $V$ bounded by the paraboloid $z=5-4x^2-4y^2$ and the $xy$-plane. The surface integral for the vector field $$\vec{F}=\bigtriangledown\times \vec{G}=2\vec{i}+2y^2\vec{j}+z\vec{k}$$ over the circle in the $xy$-plane is 15. What is…
alan.cheung
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Finding flux over a surface

If $F = 6z\mathbf i + (2x+y)\mathbf j -x\mathbf k$, evaluate $\int_S \mathbf F \cdot \mathbf n ds$ over the surface bounded by the cylinder $x^2 + z^2 = 9. x=0, y=0, z=0 $ and $ y= 6$ Okay, so I know this is a quarter cylinder in the first octant.…
Dimitri
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