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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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Application of the Gauss's Divergence Theorem

Question: Let $S$ be the closed surface forming the boundary of the region $V$ bounded by $x^2+y^2=3$, $z=0,\ z=6$. A vector field $\vec{F}$ is defined over $V$ with $\nabla.\vec{F}=2y+z+1$. What is ...
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Shortcuts for $\int_{\kappa}F dx$

Let $F(x,y,z):=\begin{pmatrix} x^{2}+5y+3yz \\ 5x +3xz -2 \\ 3xy -4z \end{pmatrix}$ and $\kappa: [0, 2\pi] \to \mathbb R^{3}, t\mapsto\begin{pmatrix}\sin t\\ \cos t \\ t \end{pmatrix}$ I was asked ...
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Surface integral of an intersection cone-plane

Find $\iint_S ydS$, where $s$ is the part of the cone $z = \sqrt{2(x^2 + y^2)}$ that lies below the plane $z = 1 + y$ The intersection of these two is an ellipse of area $A = \pi\sqrt {2}$ Note that ...
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How can we calculate the surface element of/integrate over a 2D plane in $\mathbb{R}^4$, given a parameterisation? [duplicate]

Say I have some surface: $$S=\{\phi(u,v) \mid u,v \geq 0\} \quad \text{for} \quad \phi:\mathbb{R}^2\to \mathbb{R}^4$$ Where $\phi$ is a smooth function. If I want to calculate $$\int_S f(\mathbf{x}) ...
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Volume of the fluid in the upward direction

Let $\mathbf{V}$ the velocity vector of a fluid particle at the point $(x,y,z)$ in a steady-state fluid flow. $$\mathbf{V}=x\hat{\mathbf{i}}+y\hat{\mathbf{j}}+3z\hat{\mathbf{k}}$$ Let $S$ be the ...
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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: enter link description here ...
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Calculating the area of a surface given by a set $S$

$S=\{(x,y,z):x^2+y^2+z^2=4, (x-1)^2+y^2 \leq 1 \}$. $x^2+y^2+z^2=4 \iff \frac{x^2}{4}+\frac{y^2}{4} +\frac{z^2}{4}=1$ I'm not exactly sure what to parametrize the set $S$ by I thought of using ...
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Calculate the surface integral given

Calculate the surface integral $ \ \large \int_{D} xyz dS \ $, where the surface $D$ is that part of the sphere $x^2+y^2+z^2=4$, which located above the area $y \leq x, \ y \leq 0, \ 0 \leq x^2+y^2 \...
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Flux through the surface of an ellipsoid

I was asked to calculate the flux of the field $$\mathbf A = (1/R^2)\hat r$$ where $R$ is the radius, through the surface of the ellipsoid $$\left(\frac{x^2}{a^2}\right) + \left(\frac{y^2}{b^2}\right) ...
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How can I approach these types of surface integrals, parameterization, etc

I know that it says "We prefer questions that can be answered, not just discussed.", but I'm not really sure where else I can post this question. If there's a better place or this is not allowed I ...
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Showing volume and surface integration is unaffected by the singularity at $\mathbf{r'}=\mathbf{r}$

This question is not entirely similar to the question here. Please read this question and the reader will see it is obviously not the same. $\mathbf{M'}$ is a continuous vector field in volume $V'$ ...
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How does $\cos\alpha dA=dydz$ come?

In the red rectangle, author defined what is surface integral in terms of parametric form. I am confused with the expression in the yellow rectangle. Can you please explain? How does $$\cos\alpha\; dA=...
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calculate the surface area of the part of a cylinder $ x^2 + (y-1)^2 = 1 $ that is inside the sphere $ x^2 + y^2 + z^2 = 4 $.

calculate the surface area of the part of a cylinder $ x^2 + (y-1)^2 = 1 $ that is inside the sphere $ x^2 + y^2 + z^2 = 4 $. my trial : The Domain of integration on the YZ plane is : solving : ...
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Find the Area of the orthogonal projection of $S$ on $H$ = {$3x + 2y + 100z = 0$}

Consider the surface $S$ that is the intersection of $x^2 + y^2 + z^2 = 4$ with the cylinder $(x-1)^2+y^2 \leq 1$ Find the Area of the orthogonal projection of $S$ on $H$ = {$3x + 2y + 100z = 0$} I ...
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Studying spherical coordinates

$(1)$ Please suggest some books regarding the fundamental studies on surface and volume integrals in spherical coordinates. $(2)$ Are there any books dedicated to only elementary calculus of ...
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Flux of a hemisphere

I have been asked to compute the flux through a hemisphere radius 2 centred at the origin oriented downward with $$\overrightarrow{F}=(y,-x,2z)$$ I have worked out that $$\hat{n}=-\frac{(x,y,z)}{2}$$ ...
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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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Surface area calculation of sphere segment.

I got this problem from a journal and curious how they have calculated that.Previously I asked this problem and unfortunately did not get any answer. Hereby I am posting again and hopefully someone ...
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Relationship between $(n - 1)$ forms and flux of a vector field across a hypersurface

I am currently studying about differential forms and want to deduce the Divergence Theorem (the one in $\mathbb{R}^n$) from the general Stokes' Theorem, which is obtained by taking $$\omega = \sum_{i=...
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Flux through Square on Plane

Question: Calculate the flux of the vector field $\vec{F}(x,y,z)=3\vec{i}−3\vec{j}+5\vec{k}$ through a square of side length $5$ lying in the plane $4x+2y+4z=1$,oriented away from the origin. My ...
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Surface Integral of Vector Field

Given the scalar field $$\phi(\vec{r})=\frac{1}{|\vec{r}-\vec{a}|},$$ where $\vec{a}=(-2,0,0)$, and the corresponding vector field $$\vec{F}(\vec{r})=\operatorname{grad}\phi,$$ as well as the surface $...
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Theorem of Pappus

Given a surface of revolution $S$ which can be parametrized by the map $$ \mathbf x(u,v) = (f(v)\cos u,f(v)\sin u,g(v)), $$ over the open set $U =\{(u,v) \in \mathbb R^2 \mid 0 < u < 2\pi, a <...
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Flux through sphere symmetry?

Is the flux through a sphere centered at the origin of the vector field $\boldsymbol{F} = (-x,1,z)$ equal to $0$? If so, is there any simple symmetry which suggests it? I have done the calculation ...
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Compute $\iint_S xz^2dydz+yz^2dzdx+z^3dxdy$

Problem Compute $$\displaystyle \iint_S xz^2dydz+yz^2dzdx+z^3dxdy$$ where $S$ denotes the outside surface of the common part $\Omega$ of $x^2+y^2+z^2\leq R^2$ and $x^2+y^2+z^2 \leq 2Rx$. Comment It ...
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Surface integral of a vector valued function

The value of the surface integral $$ \iint_S(x\hat{i}+y\hat{j})\cdot \hat{n}~dA $$ evaluated over the surface of a cube having sides of length $a$ is ($\hat{n}$ is unit normal vector) \begin{align*} ...
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Problem arising while calculating surface integral by taking projection.

I was asked to verify the divergence theorem for $$\vec{A}=4x\hat{i}-2y^2\hat{j}+z^2\hat{k}$$ taken over the region bounded by $$x^2+y^2=4,z=0$$ and $$z=3$$. One part ($$\iiint\nabla.\vec{A}dV$$) is ...
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How to solve the surface integral $\int\int_{S}\{(\frac{2x}{\pi}+\sin(y^2))x+(e^z-\frac{y}{\pi})y+(\frac{2z}{\pi}+\sin^2y)z\}d\sigma$?

Consider the unit sphere $S=\{(x,y,z)\in {R^3}:x^2+y^2+z^2=1\}$ and the unit normal vector $\bar{n}$ at each point $(x,y,z)$ on $S$. Then the value of the surface integral $$\int\int_{S}\{(\frac{2x}{\...
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lateral surface area of cylinder

Use cylindrical coordinates and multivariable calculus to prove that the lateral surface area of a right, circular cylinder with radius 2 and height h is 4pih. I parameterized x = rcostheta, y = ...
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parametrize the boundary of a region

I need to parametrize the boundary of this region : $D=\{y^2+z^2\le x^2+18,x^2+y^2\le 16\}$ So It's a one-sheet hyperboloid (radius=$\sqrt{18}$)+ cylinder with radius 4 I know how to parametrize ...
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How to find the flux $\int_{S} 2~dydz + dzdx + -3dxdy$ in the surface $x^2 + y^2 + z^2 +xyz = 1$ ( how to parametrize the surface ?)

Find the integral $\int_{S} 2~dydz + dzdx + -3dxdy$ where $S$ is the surface $x^2 + y^2 + z^2 +xyz = 1$ , $0 \leq x,y,z$. choose the direction of the normal as you like. i am having hard time ...
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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 calculate flux of vector field

A vector field is given as $A = (yz, xz, xy)$ through surface $x+y+z=1$ where $x,y,z \ge 0$, normal is chosen to be $\hat{n} \cdot e_z > 0$. Calculate the flux of the vector field. I tried using ...
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How to Calculate $\int_{S} xyz~d{\sigma}$ where $S$ is the portion $x+y+z=1$ in the first octant $ 0 \leq x , 0 \leq y , 0 \leq z$

Calculate $\int_{S} xyz~d{\sigma}$ where $S$ : is the portion $x+y+z=1$ in the first octant $ 0 \leq x , 0 \leq y , 0 \leq z$ . should i calculate $\sqrt{3}\int_0^1\int_0^{1-x} (xy-x^2y-xy^2)...
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calculate flux through surface

I need to calculate the flux of the vector field $\vec{F}$ through the surface $D$, where $$\vec{F} = \left<z, \, y \sqrt{x^2 + z^2}, \, -x \right> \\ D = \{x^2+6x+z^2\le 0 \,| -1\le y \le 0\}.$$...
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Help evaluating this surface integral, how to evaluate $dS$ in this?

Evaluating this surface integral $\int_{S}(a^2x^2+b^2y^2+c^2z^2)^{1/2}dS$ over the ellipsoid $S:ax^2+by^2+cz^2=1$ I am well aware of evaluating surface integral over any given sphere $x^2+y^2+z^2=a^...
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surface area with integrals

I'm working on a problem in my textbook and am confused on how to set up the integral. "Find the surface area of the part of the hyperbolic paraboloid $z= x^2 - y^2$ that lies in the first octant and ...
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How to Find the area of the portion of the sphere $ x^2 + y^2 + z^2 = 1$ between the two parallel planes .

Find the area of the portion of the sphere $ x^2 + y^2 + z^2 = 1$ between the two parallel planes $ z = a$ and $z = b$ where $-1 < a < b < 1$ are parameters. How to solve this question using ...
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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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Is my work for the following surface area problem using integrals correct?

I had to compute the surface area of portion of surface $z^2 = 2xy$ which lies above the first quadrant of X-Y planeand is cut off by the planes $x=2$ and $y=1$. Here's my work : Since, $$A(S) = ...
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Surface integral over a cone above the xy plane

From Schaum's vector analysis My approach: First, parametrize the equation $x^2 +y^2 = z^2 $ $ x = \rho cos \phi$ , $y= \rho sin\phi$ , $z= \rho$ Then, $\vec A = 4 \rho^2 cos \phi \hat i + \rho^...
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Surface integral over a cylinder bounded by 2 planes

From Schaum's vector analysis: My attempt: $\vec n = \nabla S = 2x \hat i + 2z \hat k$ $ \hat n = \frac{1}{3} x \hat i + \frac{1}{3} z \hat k $ $ \vec A . \hat n = 2xz - \frac{xz}{3} = \frac {5}{3}...
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Surface integral confusion about boundaries

From schaum's vector analysis: I project the differential area $dS$ of the plane onto the $xy$ plane, then $dxdy = dS (|\hat n. \hat k|)$ Where $\hat n$ is the normal vector to $dS$ then $dS = \...
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Surface integral of position vector over a sphere

$\iint_S$ r.n $dS$ Over the surface of the sphere with radius $a$ centered at the origin Now this is obviously trivial and the answer is $4\pi a^3$ but I want to do it the hard way because there'...
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Calculate the area of $z=\frac{x^2}{2}+\frac{y^2}{2}\;$ that is enclosed by $x^2+\frac{y^2}{4}=1$

The exercise is the text in the title. I'm studying surface integrals. To start I thougt to make a change to cartesian coordinates with $z$ as a function of $x$ and $y$, that is, $z=\frac{x^2+y^2}{2}$ ...
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Verify Stokes' theorem on a hemisphere using a line integral

When solving this question, I wrote $r(t)$ as $\langle\cos t,0,\sin t\rangle$ then I differentiated it and got $dr=\langle-\sin t,0,\cos t\rangle\,dt$, then i substituted in the formula for line ...
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surface integrals — is it “dx” or “ds”?

I was reading this note about Surface Integrals and came across this paragraph: Let S be a surface parameterized by $\mathbf X : D → \mathbf R^3$. A point $(s_0, t_0) \in D$, is mapped to $\...
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Where is my mistake? Calculating surface integral/Stoke's theorem

Let $F(x,y,z)= \begin{pmatrix} -y \\ 2x\\z \end{pmatrix}$ be a vector field and $A$ a hemisphere with $x^2+y^2+z^2=9 $ , $ z>0 $ with a circular edge at the $x,y $- level with the unit normal ...
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How to evaluate this surface integral?

I am given the following problem: Use a change of variables to evaluate $\int \int_T e^{(y-x)/(y+x)} dA$ where $T$ is the triangular region with vertices $(0,0)$, $(1,0)$ and $(0,1)$. Here is what ...
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Finding a surface integral where S is the intersection of a cylinder with a plane

If $\vec{F}=\vec{i}+2\vec{j}+\vec{k}$ and $S$ is the intersection of the solid cylinder $x^2+y^2\le1$ with the plane $2x+y-z=1$, compute $\int\int_S\vec{F}.\vec{n} dS$ (using an upward pointing $\vec{...
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Integral of function undefined at one point

Let us consider a plane with polar coordinates. Let us also consider the following integral over any area $A$ on the plane: $$\iint_A f(r,\theta)\ \hat{r}\ dr\ d\theta\ $$ Here the function is $\...