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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.

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Finding the area of the surface

FIind the area of the part of the cylinder $x^2+y^2=2ay$ that lies inside the sphere $x^2+y^2+z^2=4a^2$ The intersection of the cylinder with the sphere gives us the shape $2ay+z^2=4a^2$. So if I were to find the surface area of this shape I am…
mathnoob123
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How to evaluate $\int_0^3\int_2^0xy\,\mathrm{d}x\,\mathrm{d}y$?

The surface integral is $$\int_0^3\int_2^0xy\,\mathrm{d}x\,\mathrm{d}y$$ where $y=x$. My doubt is, which variable to integrate first $y$ or $x$? When to substitute $y=x$? I used an online integral calculator to evaluate this integral the answer…
FearlessVirgo
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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 for $y,$ but how should I find the boundaries for…
Ayoub Rossi
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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 zero. (My interpretation, not the solution). The…
Twenty-six colours
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Divergence Theorem - why can't I use it here to immediately evaluate this surface flux?

For this question I found in a video: Find the surface flux through the piece of cylinder with equation $x^2 + y^2 = a^2$ for $0 \leq z \leq h$ in the first octant, with vector field $F = (z,x,y)$. The answer says that it is $\frac{1}{2} ( ah^2 +…
Twenty-six colours
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Confused on the $dS$ component of a surface integral

I feel bad about posting an external link, but I was watching a video and the instructor parametrized a hemisphere using spherical coordinates. Now of course, if we were to transform a region of integration, we would multiply the jacobian in the…
Twenty-six colours
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Evaluating a difficult surface integral

https://i.stack.imgur.com/1HZor.jpg 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 something? Can't apply stokes theorem to…
Watch Second
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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.
Braille Espresso
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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 $r=5$? My parameterization is: $$r(u,v)=(5\cos v,5\sin…
Steve
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Surface integral of cylinder

A cylinder surface in $(x,y,z)$-space is given by the parametric form: $$ \begin{bmatrix} x \\ y \\ z \end{bmatrix}=r(u,v)=\begin{bmatrix} \exp(u)+\exp(-u) \\ 2u \\ v(\exp(u)-\exp(-u)) \end{bmatrix}, 0 \leq u \leq 1, 0 \leq v…
Steve
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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 point set $B$. I'm not completely sure how to do this. Do…
Steve
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Finding the limit of surface integral

I have found this problem in my textbook. Evaluate $$\iint_S F\cdot n ds $$ where $$F = yz\hat i+xz\hat j+xy\hat k$$ and S is that part of the surface of the sphere $$x^2+y^2+z^2 = 1$$ which lies in first octant. I have tried to solve this…
IAmBlake
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Criteria for a surface to have an empty boundary?

If we consider an orientable and regular surface $\Sigma \subset \Bbb R^3$, as we would consider in Stokes' theorem, do we have any criteria that would guarantee its boundary to be empty ? For example, I know that the sphere $\Sigma_1 = \{(x, y, z)…
Desura
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Write double integral from D

$D$ - sector in picture I solve it and want know if it's right solution. My solution $D = \{-y \leq x \leq y, -1 \leq y \leq 1 \}$, so $\iint\limits_{D}f(x, y) = \int\limits_{-1}^{1}dy \int\limits_{-y}^{y}f(x, y)dx$
Stepan Vanzuriak
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Triple integrals-finding the volume of cylinder.

Find the volume of cylinder with base as the disk of unit radius in the $xy$ plane centered at $(1,1,0)$ and the top being the surface $$z=((x-1)^2+(y-1)^2)^{3/2}.$$ I just knew that this problem uses triple integral concept but dont know how to…
user371841
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