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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Evaluating Surface Integral with Divergence Theorem

Evaluate $\displaystyle\int_S \mathbf{F\cdot n}\ dS$ over the entire surface of the region above the $xy$ plane bounded by the cone $z^2=x^2+y^2$ and the plane $z=4$ if $\mathbf F=x\hat i+y\hat j+z^2\...
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How to find the surface integral of $f(x,y,z)=e^{-z}$

How can I find the surface integral of $f(x,y,z)=e^{-z}$ given that $x^2+y^2=9$ and $0\leq z \leq 2$. I started by changing the $x$ and $y$ coordinates to polar coordinates: $x=3\cos(\theta),\ y=3\sin(...
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Flux of two open surfaces sharing the same boundary in a solenoidal field

I am studying for exam and I come across a dilemma. Assume that: $$ \vec{F}=\vec{\nabla}\times\vec{A} $$ This implies that we are in a solenoidal or divergence-less vector field. It is given that $\...
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2 answers
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Surface integral calculation doubts

I have vector field $\vec{F}=\frac{\vec{r}}{\lvert \vec{r} \rvert^2}=\frac{\vec{r}}{r^2}$ and I want to calculate net flux from a sphere of radius $R$ using spherical coordinates. This is what I tried ...
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What am I doing wrong with this surface integral?

I'm asked to integrate $\vec{A}=r\cos\theta \ \vec{u_r}+r\sin\theta\ \vec{u_\theta}+r\sin\theta\cos\varphi \ \vec{u_\varphi}$ over a hemisphere of radius $a$. I take $d\vec{S}=r^2\sin\theta d\theta d\...
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3 votes
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Parametrize ellipse in the xy-plane

I want to parametrize the surface $(\frac{x}{a})^2 + (\frac{y}{b})^2 = 1$ in the xy-plane in $\mathbb R^3$ My attempt is $G(r,\theta) = (r \cos\theta,\frac{b}{a} \sin \theta,0)$ where $ \theta \in [0,...
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Verify Stokes' theorem: ∬ (curlF) • n dA = ∮ F r’(s) ds When F=yi +zj +xk and S is a paraboloid z=f(x,y)=1-(x^2–y^2 ), z≥0

I found out the curl f which came to be -i-j-k. After that I am stuck on what to do with paraboloid and the limits to be used on both sides to verify the theorem. Please can someone share the detailed ...
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Find the area of the surface generated by revolving the curve $g(y) = 4√y$ from (4, 1) to (12, 9) about the y-axis

Find the area of the surface generated by revolving the curve $$g(y) = 4√y$$ from (4, 1) to (12, 9) about the y-axis. So far what I've figured out is setting up the formula: $$\int_{1}^{9}2\pi x \sqrt{...
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Integral of Indicator function on surface with Minkowski content

I saw on this wikipedia page the following expression: $$-\int_{\mathbb{R}^{n}} g(\mathbf{x}) \frac{\partial 1_{D}(\mathbf{x})}{\partial n} d \mathbf{x}=\int_{S} g(\mathbf{s}) d \sigma(\mathbf{s})$$ ...
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Average value of $|x_1|$ over $n$-sphere

Suppose $(x_1,\ldots,x_{n+1})$ are coordinates of points on n-sphere. What's the average value of $|x_1|$? Let $f(n)$ be such value for $n$-sphere. Mathematica suggests that $$f(10)=\frac{63}{256}$$ ...
1 vote
1 answer
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How do I compute the area of the surface where $z=x^2+y^2$ and $0\leq z\leq 1$?

Let me define the surface where $z=x^2+y^2$ and $0\leq z\leq 1$. I want to compute the area of this surface. My idea was to parametrize the surface with the following parametrization $$\phi(r,\theta)=...
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Surface area of inverse functions

I was wondering if the area of a curve rotated about the x-axis could be found by rotating the inverse curve about the y-axis because I have to find the the area by rotating $x = \frac{1}{4} y^2 - \...
2 votes
1 answer
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surface of $M=\{(x,y,z)\in\mathbb R^3: x^2+y^2<1 \text{ and } z=3-2x-2y\}$

For $M=\{(x,y,z)\in\mathbb R^3: x^2+y^2<1 \text{ and } z=3-2x-2y\}$ I want to determine the surface of $M$, this means I want to calculate $\int_M 1 dS_M$. I don't know how to this and I only need ...
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Normal vector of a surface in $\mathbb{R}^n$

Studying surface integrals I have come up with a problem when I try to generalize some formulas. In case I need a normal unit vector of a surface in $\mathbb{R}^n$, how can I get it? For a surface in $...
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Aerodynamics Question: Calculating force and moments of a surface using C_p at various points.

I have been given a data that contains x, y, z coordinates; coefficient of pressure and the coefficient of friction at different points. So far, I tried calculating the force coefficients and moment ...
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How to determine positive orientation for the purposes of Stokes Theorem

I want to determine the positive orientation in the case where curve $C$ is given by the intersection of sphere $x^2+y^2+z^2-2ax-2ay=0$ and plane $x+y=2a$ starting from the point $(2a,0,0)$ and going ...
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Integral of quaternion function involving Dirac operator

Let $\Omega$ be a bounded set in $\mathbb{R}^3$ and $f,g:\mathbb{R}^3\to\mathbb{H}$ are two quaternion-valued function \begin{align*} f(x) &= f_0(x) + if_1(x) + jf_2(x) + kf_3(x) \\ g(x) &...
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Surface integral of $f(\vec x) = z^2$

Let $\vec x = (x,y,z)^T$ and $f(\vec x) = z^2$. Calculate the surface integral $$ \int_{|\vec x| = \frac{1}{2}} f \, \, dO $$ my solution : $dO = r^2 sin(u) \, \, du \, dv$ and $f(\vec x) = z^2 = r^2 ...
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How do you calculate the fluid flowing through a pipe using a flux integral?

I am beginning to learn vector calculus and over the last few days I have been looking at integrals. An example the lecturer used was fluid flow. He said that the volume of fluid flowing through a ...
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Estimate flux integral of $\frac{1}{|x|}$ in $\Bbb{R}^3$ through a quadratic surface.

My question occured reading this paper in the first example of section 6. Let $f:\Bbb{R}^3 \rightarrow \Bbb{R}^3$, $f(x)=|x|$ a vector field. Now divide $\Bbb{R}^3$ up into disjoint unit cubes ...
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Surface integral ( limits of surface)

Calculate $$\iint_{S} \frac{xy}{\sqrt{1+2x^{2}}} dS$$ where $$S =\{(x, y, x^{2}+y ) , 0 \leq x \leq y, x+y \leq 1\} $$ My attempt is this i cannot figure out the limits of $x$ and $y$ In the ans it ...
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Surface integral of a sphere via definition

I attempted to calculate the surface integral $\iint_S F \cdot n \ dS$ where $F = (x^2 + y^2 + z^2)(x, y, z)$ and $S$ is the sphere of radius $a$ centered at the origin. I used the divergence theorem ...
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computing the flux integral on this surface

Compute the flux integral for the field $\mathbf{F} = x\mathbf{i} + y\mathbf{j} + 5\mathbf{k}$, where $S$ is the boundary of the cylinder enclosed by $x^2+z^2=1$ and the planes $y=0$ and $x+y=2$. How ...
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Flux of $\mathbf{F}=(xz,yz,x^2+y^2)$ across paraboloid $z=4-x^2-y^2,\ z\geq 0$

I have calculated the flux of the vector field $\mathbf{F}=\begin{bmatrix}xz\\ yz\\ x^2+y^2\end{bmatrix}$ outward across the surface of the paraboloid $S$ given by $z=4-x^2-y^2,\ z\geq 0$ (with ...
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4 votes
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How to evaluate double integrals of a surface over a specific region?

I found this exercise while exercising for the exam: Let $T$ $\subset$ $R^2$ be the triangle with these vertices $(0,0), (2,0), (0,1)$ and let $\Omega$ be the surface defined like this: $\Omega$ = {$(...
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1 answer
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Computing surface area of a plane

I am learning about surface integrals and flux integrals and I have done the following exercise for practice. Since I am still not entirely sure if I have understood correctly I would appreciate some ...
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How to solve a double integration with two different region that have different bound?

The height of the equation is $z= x^3 +4y$ and it is bounded by $Y = x^3$ and $Y = 2x$, I have drawn the bounded region below. As you can see, for the left region if we take the double integration of ...
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Unit vectors in line and surface integrals

I have been reviewing coordinate systems and this image is an out take from my lecture notes. This is in cylindrical coordinates which I understand however, what I don't understand is why the line and ...
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Can I calculate the area of a closed surface on a sphere by its boundary using Stokes' Theorem?

So I (just learned that I) can get the area of any closed area $S$ with boundary $\partial S$ on the flat plane through Stokes' (or in the simplified case Green's) theorem by using a vector field with ...
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Integral of a function over the 2-sphere (Gauss theorem-explanation needed)

Let $\mathcal S^2:=\{(x,y,z)\in \mathbb R^3 \vert\; x^2+y^2+z^2=1\}$ be the standard 2-sphere. For some $p=(x,y,z)\in \mathcal S^2$, the outer unit normal field of $B(0, 1)$ on $\mathcal S^2$ is given ...
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Area of a sphere via surface integral

As an example for the computation of the area of a surface of rotation with surface integrals, my textbook derives the formula for the area of a sphere of radius $R$. The sphere is described by $x^2+y^...
1 vote
1 answer
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A Question about evaluating surface integrals

Vector calculus is relatively new to me so I have a little trouble understanding double integrals intuitively. The question is esssentially this: Calculate the surface integral of $\vec v=x^2 \hat j $...
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Prove $\frac{1}{v(B_{\varepsilon})}\iint_{S_{\varepsilon}}zf{\rm d}S\underset{\varepsilon\rightarrow0}{\rightarrow}\frac{\partial f}{\partial z}$

Let $f:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be a continuously differentiable function. Let $S_\varepsilon,B_\varepsilon$ be the sphere and ball of radius $\varepsilon$ around the origin, respectively. ...
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Volume of a solid of revolution: check and information

I have to calculate the volume of the solid of revolution around the $Y$ axis generated from the curve $f(x) = 2x^2-4$ between $[0, 2]$. So I used the definition: $$V_y = 2\pi\int_a^b x f(x)\ \text{d}...
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Without Green’s theorem, why is $\oint_C(xdy) = \Delta A$

A textbook that I am reading states that $\oint_C(xdy) = \Delta A$ and I do not understand why. Any other place on the internet uses Green’s theorem, but the book is using this identity to derive the ...
1 vote
2 answers
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Surface Integrals for Calculating Volume

I understand that volume under a surface can be calculated with double integrals in Multivariable Calculus, but can it also be calculated with surface integrals? I would think that taking the surface ...
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Flux through surface of revolution

I'm trying to solve the following problem Let $C$ be the curve in the $xy$ plane given in polar coordinates by $r = 2-\sin(\theta),\ 0 \leq \theta \leq \pi$ and let $S$ be the surface given by ...
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Find volume of solid bounded by $z=x+y,(x^2+y^2)^2=2xy,z=0,(x>0,y>0)$

Moving into polar coordinates find volume of solid bounded by given surfaces. $z=x+y,(x^2+y^2)^2=2xy,z=0,(x>0,y>0)$ Moving into polar coordinates we get. $z=r(cos\phi+sin\phi),r^2=sin(2\phi),z=0$...
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Find volume of solid bounded by $x^2+y^2=R^2,x^2+z^2=R^2$ [closed]

Find volume of solid bounded by given surfaces. $x^2+y^2=R^2,x^2+z^2=R^2$ I am thinking finding one region above $xy$ plane and multiply by $4$. $ V = 4 \times \int_{-R}^R \int_0 ^ \sqrt{R^2-x^2} x^2+...
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Parametrizing Surface Area Correctly

If the surface S is the area that between cylinder $x^2+y^2=1$ and $z=0$, $z=x+2$. Then calculate the surface integral: $$\iint\limits_{S} xz\cdot \mathrm{dS}$$ Here the plotting: I parametrized by: ...
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Can $A(S^-)>A(S^+)$?

Let's be $S$ a surface which is homeomorphic to a $\mathbb{S}^2$ (the unit sphere). Let's be $$S^+=\{p\in S : K(p) \geq 0\} \text{ and } S^-=\{p\in S : K(p) \leq 0\}$$ being $K$ Gaussian curvature. Is ...
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Volume of a square pyramid with a curved base

Consider a square pyramid of base length $a$ and height $H$ with vertices at coordinates $\left(\pm \frac{a}2, \pm \frac{a}2\right)$ and $(0,0,H)$. Assume that $f : \Bbb{R}^2 \to \langle 0,H\rangle$ ...
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Surface integral of hemisphere

In a scalar field I need to calculate the surface integral of this: $$\iint_{\Sigma}\frac{d \sigma}{\sqrt{x^2+y^2+(z+R)^2}}$$ with $\Sigma$ the upper half of the sphere $x^2+y^2+z^2=R^2$ The formula ...
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Area of part $S$ of the sphere $x^2+y^2+z^2=a^2$ that lies inside the cylinder $x^2 + y^2 = ax$ [duplicate]

By symmetry, I can calculate as $$2\int_0^{\pi/2} \int_0^{a\cos\theta} \frac a{\sqrt{a^2-r^2}} r\,dr\,d\theta= a^2(\pi-2)$$ but when I calculate as $$\int_{-\pi/2}^{\pi/2} \int_0^{a\cos\theta} \frac a{...
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Divergence Theorem over a Simple Smooth Surface and at its Boundary

I have to solve the following theoretical problem: In $\mathbb{R}^3$ with a fixed Cartesian coordinate system $(x,y,z)$, let $D$ be an open subset in the $(x,y)$ plane. $D$ can be considered as a ...
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Volume to surface integral of $R\times \nabla \times B$

I need to transform the following integral into a surface integral (if that's possible): $$\int\int\int_\Omega R\times (\nabla \times A) dv = \int\int_{\partial \Omega} ? . {\bf n} da, $$ where $R = (...
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Surface integral of vector field $\vec{F}=\langle 2x^3,5xz^2,x^2+6y^2z \rangle$

I want to evaluate the surface intgeral of $$\vec{F}=\langle 2x^3,5xz^2,x^2+6y^2z\rangle$$ in the $xy$-plane and $x^2+y^2\le4$. I know I should use $$\iint_S\vec F\cdot d\vec{S}=\iint_S\vec F\cdot \...
2 votes
0 answers
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Surface Integral of Vector Field $\textbf{A}=(xz,0,yz)$

I have to solve the following problem: A cylinder of radius R in $\mathbb{R}^3$ with axis along the $z$-axis of a Cartesian coordinate system $(x,y,z)$ can be parametrised as $\textbf{x}(\theta,z) = (...
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Integral with respect to other parametrization.

I'm studying Do Carmo's differential geometry. Let $Q$ be a compact region in $\mathbb{R^2}$ that is contained in a coordinate nbd $X : U \rightarrow S$, where $S$ is regular surface. I want to get ...
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Surface integration over a rotated rectangle

Computing the area of, or integrating over, a rectangle aligned with the x and y axes is really straightforward. Indeed, if $x \in [-\frac{b}{2},\frac{b}{2}]$ and $y \in [-\frac{y}{2},\frac{y}{2}]$, ...

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