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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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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)}{...
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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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107 views

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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1answer
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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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31 views

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 $\...
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119 views

Volume between cone and sphere of radius $\sqrt2$ with surface integral

Consider the cone $z^2=x^2+y^2$ between $z=0$ and $z=1$. Find the volume of the region above this cone and inside the sphere of radius $\sqrt2$ centered at the origin that encloses the cone. The ...
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surface area of two connected surfaces

If you want to compute the surface area bounded by the upper hemisphere and the paraboloid, do you have to split the integral into two different surface integrals ?
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Stokes' Theorem - Vector Field

I am having problems trying to verify Stokes' theorem (below) as part of a question. $$\iint_{S} \text{curl} \vec F \cdot d\vec S=\oint_{c} \vec F \cdot d\vec r$$ The vector field in question is $\...
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Surface integral of curl

Let $\vec F(x,y,z)=(x^2+y^2+\sin(xy)$, $e^x$+$2xy, -yz$). Calculate the flow of $\nabla\times\vec F$ through the following surface: $$ S=\{(x,y,z) x^2 + 2y^2 + 3z^2=10, y\ge 0\} $$ I figured that if ...
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Calculate the flow of $\vec{F}(x,y,z) = 3x\vec{i} + 3y\vec{j} + z^5\vec{k}$ over the surface $x^2 + y^2 = 25$ for $0 \leq z \leq 1$

Calculate the flow of $\vec{F}(x,y,z) = 3x\vec{i} + 3y\vec{j} + z^5\vec{k}$ over the surface $x^2 + y^2 = 25$ for $0 \leq z \leq 1$. The normal considered points inside. The book uses cylindrical ...
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67 views

How to show $\unicode{x222F} \dfrac{\hat{r} \times \vec{dS}}{r^2}=0$

I can do the following derivation using solid angle: $$\unicode{x222F} \dfrac{\hat{r} \cdot \vec{dS}}{r^2} =\unicode{x222F} \dfrac{dS \cos\alpha}{r^2} =\int^{2\pi}_0 \int^\pi_0 \sin\theta\ d\theta\ d\...
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1answer
86 views

calculate the surface integral in the upper hemisphere

Calculate the surface integral $f(x,y,z)=x^2+y^2+z^2$ in the upper hemisphere of the sphere $x^2+y^2+(z-1)^2=1$ I tried to compute the value of the surface integral $\iint_S{F.n} dS$ with the ...
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1answer
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Surface area of $x^2+z^2=a^2$ inside of $x^2+y^2 = 2ay$ and in first octant

The questions is What is the surface area of $x^2+z^2=a^2$ inside of $x^2+y^2 = 2ay$ and in first octant? My attempt The second equation can be rewritten as $x^2 + (y-a)^2=a^2$ to make it easier ...
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1answer
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Surface integral for area calculation

Is my procedure correct? Calculate paraboloid area portion of equations $$P \equiv (u \cos v, u \sin v, u^2) $$ with $0 \leq v \leq \dfrac{\pi}{4}$ and $0 \leq u \leq \dfrac{1}{2}\tan v$ ...
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1answer
134 views

How to evaluate a parameterized surface integral?

Suppose you have to evaluate the surface integral $$\int\int_S (x^2+y^2+4)\space dS$$ where $S$ is the surface parameterized by $\textbf{r} = <2uv, u^2-v^2, u^2+v^2>$ with $u^2+v^2 \le 16.$ I ...
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1answer
42 views

Calculate integral where S is the surface of the half ball $x^2 + y^2 + z^2 = 1, \space z \geq 0,$ and $F = (x + 3y^5)i + (y + 10xz)j + (z - xy)k$

I am asked to calculate this integral and want to make sure I am doing this set up correctly, So I tried to paramatize this surface by : $x = rcos(\theta), \space y = rsin(\theta), \space z = \sqrt{1 ...