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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.

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Integrating with respect to surface measure

How does one integrate with respect to a surface measure, is it the same as a surface integral? I've tried a google search but couldn't find much. In particular I have the following problem but can't ...
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Can someone check if my proof is correct?

I was working a tutorial and it had this proof listed below. It says that S is a closed surface and H is a region $$\int_S \frac{\textbf{r.n}}{r^2} dS\, = \int_H \frac{dH}{r^2} \,$$ My approach ...
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Surface integral - cone below plane

After several years I suddenly need to brush up on surface integrals. Looking through my old Calculus book I have been attempting to solve some problems, but the following problem has really made me ...
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35 views

Find the flow over a vector field through a sphere

Assuming that we can use the Gauss theorem and given $$\vec f(x,y,z)\quad=\quad\left(\;g(z/y),\;g(x/z),\;3z+g(y/x)\;\right)$$ find the flow over $\vec f$ through the surface $$(x-2)^2+(y-2)^2+(z-2)^2\...
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Gradient vector $\nabla F = (z_x, z_y, -1)$ is normal to the integral surface?

I have an integral surface $z = z(x, y)$. Writing this integral surface in implicit form, we get $$F(x, y, z) = z(x, y) - z = 0$$ I am then told that the gradient vector $\nabla F = (z_x, z_y, -1)$ ...
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Area enclosed between two curves

Find the area enclosed by the curve in which the plane $z=2$ cuts the ellipsoid $x^2/25 + y^2 + z^2/5 =1$ I tried to solve this by projecting the area on $xOy$ plane and I got the final answer as $\...
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How to find the surface area of the n-1 sphere?

Q: Pick n>0 and suppose that Vol_n is the volume of the n-ball B(0,1), find the surface area of the n-1 sphere ∂B(ξ,ρ)? I am totally stucked at where to start this problem. I have tried the ...
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Volume integral sphere [closed]

Compute $$\iiint_E \frac{ \,dx \,dy \,dz}{\sqrt{x^2+ y^2 +(z-2)^2}}$$ where $E$, i.e. the domain of integration, is specified by $$x^2+y^2+z^2 \le 1$$ I tried using spherical coordinates But i ...
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What is the first moment of the image irradiance?

Recently I took on the task of surface reconstruction. And now I try to take a grasp on how to find the illumination direction given gray-scale image under Lambertian assumption. The formula I stuck ...
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Spherical coordinates in surface integrals

I am stuck on the following problem Evaluate : $$I=\iint x^2 y^2 z dS $$ where S is the positive side of lower half of the sphere $x^2 + y^2 + z^2 = a^2$ I tried using spherical coordinates and ...
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38 views

Find area of a curved surface within a condition

Curved surface r given by $\mathbf{r}=(u+v,u-v,uv)^T$. Calculate area of a part of the curved surface that satisfies $u^2 + v^2 \leq 1$ Here what I have done: $\mathbf{r}_u=\frac{\partial \mathbf{...
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Theorem about integration on surfaces

Given two k-surfaces $\sigma:[-a, a]^k\rightarrow\mathbb{R}^n$ and $\sigma':[-b, b]^k\rightarrow\mathbb{R}^n$, if $\sigma([-a, a]^k) =\sigma'([-b, b]^k)$ and $\exists\, g\, :\sigma'=\sigma\circ g$, ...
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Finding $|r_u \times r_v|$ for the Surface Integral and a Flux question.

So I am going over Surface Integrals right now, and I have a question about certain calculations. We know that $\iint_SdS=\iint_D |r_u \times r_v| \,dA$ if $r(u,v)$ is the vector function that ...
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Why does the gradient of a level surface represent the differential tangent plane?

I understand that the gradient is the direction fastest rate of change and why this is true, but just because its direction is orthogonal to the surface, doesn't mean its magnitude is that of the ...
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Computing $\int_{S} \vec{F}.\hat{n}\,dS$ where $S:\{(x,y,z)\in R^3:x^2+y^2+2z=2,z\geq 0\}$

Let S be the surface $\{(x,y,z)\in R^3:x^2+y^2+2z=2,z\geq 0\}$ and let $\hat{n}$ be the outwards unit normal to S. If $\vec{F}=\langle y,xz,x^2+y^2\rangle$ then find the value of $\int_{S} \vec{F}.\...
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Line integral with z coordinate varying

I have to calculate the area of the surface $$ S=\{(x,y,z) \in \mathbb{R^3} :x^2+y^2 = 2, 0 \le z \le 5-2x\}$$ If I parametrize $r(t)=(\sqrt2 \cos t, \sqrt2 \sin t)$ then I cannot say that $z=f(x,y)=...
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Integrating vector fields over higher-dimensional surfaces

I know how to integrate both scalar and vector functions over 2-dimensional surfaces. The way to do that is the same explained here https://en.wikipedia.org/wiki/Surface_integral The thing is that it ...
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How to find the projected area in the x-z plane of an ellipsoidal cap rotated by angle β in x-y plane?

I have ellipsoidal cap rotated in the x-y plane by an angle $\beta$; where the axis size in x coordinate is 'a', the axis size in y-coordinate is 'b' and axis size in z coordinate is 'c'. I am trying ...
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Surface area of curved figure [closed]

SOLVED I have a task: calculate surface area. There is given next expressions: $$z=x^2+y^2\quad z^2\leq xy\quad x\geq 0 \quad y\geq 0$$ This look like this from above: view from above And like this ...
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Calculate $\int_S \vec{A} \cdot \hat{n} \ dS, S=\{x^2+y^2+z^2=R^2, z \ge 0\}$

Calculate $$\int_S \vec{A} \cdot \hat{n} \ dS$$ $$\vec{A}=4 \hat{i}$$ Using spherical coordinates: \begin{cases} \hat{r}=\sin \theta \cos \phi \ \hat{i} + \sin \theta \sin \phi \ \hat{j}+ \cos \...
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Question on the definition of surface integrals

For a surface S given explicitly as the graph of z = ƒ(x, y), where ƒ is a continuously differentiable function over a region R in the xy-plane, the surface integral of the continuous function G over ...
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Finding latitude in a tilted coordinate system

Suppose I have a sphere where $\varphi = 0$ is vertical, so I can have lines of latitude where $\varphi = \frac{\pi}{2}$ is pointing to the "equator". My question is, suppose I tilt my coordinate ...
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Area of surface cut off a sphere by a cone

Find the area of the part of the sphere $x^2+y^2+z^2=4$ cut off by the cone $z = \sqrt{x^2+y^2}$ First of all, I want to set up the integral only on the part of the sphere that lies inside the cone. ...
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167 views

Find the surface area of the part of the sphere $x^2 + y^2 + z^2 = 16$ inside the cylinder $x^2 - 4x + y^2 = 0$

Find the surface area of the part of the sphere $x^2 + y^2 + z^2 = 16$ inside the cylinder $x^2 - 4x + y^2 = 0$ First of all, I need to find the equation of the plane along which these two solids ...
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1answer
34 views

Finding the normal vector of surfaces

I'm learning about surface integrals, and while I know how to find the curl of a vector field and how to simplify the dS, I have trouble finding and understanding how the normal vector of a surface is ...
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Finding surface of the set in $\mathbb{R}^3$

Given set $$M \equiv x^2 + y^2 \leq 2z,~z \in [0, 1]$$ find it's surface. Using cylindrical coordinates I'm finding that $$r^2 = 2z \implies z = \frac{r^2}{2}.$$ Now my transformation is of the form ...
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Gauss' law and a half-cylinder

The question is: A half cylinder with the square part on the $xy$-plane, and the length $h$ parallel to the $x$-axis. The position of the center of the square part on the $xy$-plane is $(x,y)=(0,...
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2answers
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Intersection of a sphere and a plane.

I have a curve: $\gamma =:[ x^2+y^2+z^2 = 1, x+y+z=1] $ How I can parametrize it? Or write it out in explicit form? I need this for compute the integral $\iint_{S}dS$, where $S$ is a surface of ...
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How to do this surface integral?

I first parameterise the surface as $\vec{r}(x,y)=(3x,2y,6-3x-2y)$ where $ 0 \le x \le 2 $ and $ 0 \le y \le (6-3x)/2 $. And I am stuck here, don't know how to find the correct ${T_x} \times {T_y} $(...
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Computing $\iint_B(x^2+2y^2-3z^2 )\,dS$ over unit sphere $B\subset\mathbb R^3$

Let $B$ be the unite sphere in $R^3$. Then what is the value of $$\iint_B(x^2+2y^2-3z^2 )\,dS$$ over the surface $B$ ? I substituted the value of $z^2$ as $1-x^2-y^2$ and then integrated the ...
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Volume Loss Rate

A spherical hot-air balloon with radius $R$ lies on the $(x_1,x_2)$-Plane. The balloon has an opening at the bottom right where the radius is $\frac{R}{4}$. The center of the opening is at the origin....
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Surface comprising the union of two smooth surfaces?

I have a question at the moment which states that S is a solid spherical shell bounded by the surface H, comprising the union of two surfaces (which are spheres) which are: $x^{2}+y^{2}+z^{2}=1$ $x^{...
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3answers
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Computing the surface integral of the octant of a sphere with polar coordinate substitution

Let me first describe where I start: $$\iint_Sz^2\,dS$$ We want to compute the surface integral of the octant of a sphere $S$. The radius = 1. The sphere is centered at the origin. $$S=x^2+y^2+z^...
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1answer
61 views

Flux across elliptic cylinder

The vectorfield $F$ is given by $F(x,y,z) = [x^2y+z^2, xcoz(z)-xy^2, x^3+3z]$. Let $S$ be the cylindric surface defined by $x^2+4y^2=1$ where $0 \leq z \leq 8$. Calculate $$\iint_S F \cdot \hat{N} ...
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Faster way to calculate the area of this surface

I have to solve this exercise: Find the area of the portion of the surface $z=xy$ included between the two cylinders $x^2+y^2=1$ and $x^2+y^2=4$ what i did so far: I parameterized the surface ...
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1answer
32 views

Volume / Surface of a Paraboloid through Integration

Given is a Paraboloid delimited as following: $$z_1 = a(x^2 + y^2),\ z_2 = h $$ That's my try for the Volume computation: First I find the radius of the circle resulting from the intersection ...
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What's wrong way the way that I parametrized the torus

I know the standard way to parametrize a torus given by Wikipedia. However, I'm trying to parametrize a specific torus, where $R = 2$ and $r = 1$ in Wikipedia's notation by doing this: (Note that my $...
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Double / Surface Integral over a circular area

To compute is the following integral: $$\int_{A} dx \ dy \ (x + y)^2$$ over the area $A$, which is the circular area between: $r_0 ^2 < x^2 + y^2 < r_1^2$ My approach: 1) Parametrize: $$x( ...
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1answer
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Surface area of ​the figure of rotation

For example, I want to rotate a curve $z = x^3$ around the axis $OZ$. I have a surface: $z = x^3 + y^3 $. Now I want to compute a area of this surface. Parametrize it: $ x = rcos\phi \\ y=rsin\phi \\...
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My Generalized Mean Value Theorem

I'm trying to prove the following without the divergence theorem(for soon to be obvious reasons): Let $F: \mathbb{R^3} \to \mathbb{R^3}$ be a differentiable function. In $\mathbb{R}^3$, let $V$ be a ...
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1answer
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Parametrization of parabolic hyperboloid.

Please, help me! How I can parametrize this surface: I have a parabolic hyperboloid. $ H := (z = xy) $ intersected by a cylinder whose base is a unit circle centered at the point $(0, 0, 1)$ I tried ...
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1answer
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Surface area of paraboloide inside sphere

Find the surface area of the part of the paraboloide $z=\frac{x^2+y^2}{2}$ inside the sphere $x^2+y^2+z^2=3$ Setting $2z = x^2+y^2$ I obtain that the points of intersection between the paraboloide ...
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1answer
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Multivariable Calculus, Help with pre-Bachelos Homework [closed]

I need help with a couple of problems for homework, btw I need to finish this in less than 12 hours, I'll appreciate your help, and srry if something is wrong or is confusing I'm not native speaker. ...
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1answer
100 views

Surface integral over a sphere - with strange limits

I've looked at tons of videos and read all chapters in my book and can't seem to be able to solve this task. Evaluate the surface integral $$\int\int_{Y} (x+y)z dS$$ where the surface Y is the part ...
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28 views

Finding a specific surface integral

Calculate $$ \iint_S Z ds $$ where $S$ is the surface whose sides $S_1$ are given by cylinder $x^2+y^2=1$, whose bottom $S_2$ is the disk $x^2+y^2$ is less than or equal to $1$ in the plane $z=0$, ...
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1answer
37 views

Integrating a function over a region

I have the following problem on a homework reading and am unsure how to attempt to solve it: Integrate $f(x,y) = x^2\sin(y)$ over the graph of $g(x,y) = 2x-2y$, on the domain $[0,1]$x$[0,\pi]$. The ...
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1answer
164 views

Integral over a surface in 4-dimensions

Consider the integral of a function $f(x,y,z)$ over a surface embedded in 3 dimensions. The surface has a parameterization: $$g(u,v) = (x(u,v), y(u,v), z(u,v)) $$ The integral is given by: $$ \iint_{...
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1answer
46 views

Evaluate Surface integral: $\vec F=y{\hat i}+x{\hat j}+zy{\hat k}$, S is part of $z=x^2+y^2$ above $z=1$. Assume S has an upwards orientation.

I want to evaluate ${\int}{\int_S}{\vec F}\cdot d{\vec S}$ with the given information in the title but cannot for the life of me figure this one out. I have looked at Stoke's theorem, and also the ...
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148 views

Calculate the flux through cylinder $x^2+y^2 \leq 9$

So the question is Calculate the flux $F(x,y,z)=(x,y,xz)$ through the surface $$S= \{(x,y,z) \in \mathbb{R}^3: x^2+y^2=9, 0<x<3, 0<y<3, 0<z<9\}$$ away from the z-axis. I ...
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0answers
27 views

computing fairly complicated integrals on the sphere

Assume $D$ is a diagonal matrix $2n\times 2n$. Let $J$ be the standard symplectic matrix, i.e.: $$J:=\begin{pmatrix} 0 &-\text{id}\\ \text{id} &0\end{pmatrix}.$$ I would like to compute the ...