1
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1answer
47 views

Notation for Surface Integral in $\mathbb{R}^3$

Recently, a paper of mine got accepted, but the reviewers are struggling with the (in my view) standard notation for surface integrals in $\mathbb{R}^3$: Let $\Gamma \subset \mathbb{R}^3$ be a ...
0
votes
1answer
76 views

Surface Integral calc 3

I am having difficulty setting up this problem. I know the bounds must be 0 to pi/2 for both theta and phi but I am unsure as to how to calculate the integrand. I know it must be the double integral ...
1
vote
1answer
89 views

How do you find the surface area of a boundary in R3?

I need to solve this problem: Let $D=\{(x,y,z):4(x-2+z)^2+4y^2\le(2-z)^2,0\le x-z\le1\}$ Calculate the area of $\partial D$ So how do you calculate the area of the boundary of a volume defined ...
1
vote
4answers
37 views

Help calculating the surface area given by the polar curve: $r=2(1-\cos\theta)$

I want to calculate the surface area given by the curve: $$ r = 2(1-\cos(\theta)) $$ using an integral. I have thought about doing this: $$ x = r\cos(\theta), \, y = r\sin(\theta) $$ $$ \iint r \,dr ...
2
votes
1answer
27 views

Newtonian potential at (0, 0, – a)

I found this problem in the book Advanced Calculus, written by Friedman. "Newtonian potential at (0, 0, – a) due to a mass with constant densinty $\sigma$ on the hemisphere S: $x^2 + y^2 + z^2 = ...
1
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2answers
26 views

Surface: intersection of 2 polar curves

I have these two polar curves: $$ C_1: r = 2 - \cos(\theta)\\ C_2: r = 3 \cos(\theta) $$ Plots: C1 and C2. I need to find the surface of $D = C_1 \cap C_2$. I started by finding the solution to ...
0
votes
0answers
13 views

Scalar potential, vector field and line integral

I've been trying to get my head around this topic for a project where I need to reconstruct a 3D surface given the estimated normals of it (Photometric Stereo). I just want to be sure I'm ...
1
vote
1answer
34 views

Questions about surface integrals and an example problem

It is- Double integral(x + y) dS Where S is the part of the cylinder y^2 + z^2 = 4 . With x being between 0 and 5 First question, if we want to get the integral of the surface of a cylinder, I ...
1
vote
1answer
11 views

Double integrals using cylindrical coordinates

Suppose we had a vector field $F = (axy^2,ayx^2,x^2\cos(\pi z))$ and wanted to calculate the surface integral of $$\int\int_SF\cdot n \ dS$$ where $S = \{(x,y,z): x^2 +y^2 = 1, 0 \leq z \leq 1/2 \}$ ...
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votes
1answer
22 views

Parametrize plane and get surface area

Find a parametrization of the surface: $y + 2z = 2$ inside the cylinder $x^2 + y^2 = 1$. Then, compute its surface area. I'm having trouble finding the parametrization of the surface. I don't think ...
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votes
0answers
43 views

Finding surface integral of the paraboloid and disk

Let S be the surface consisting of the paraboloid $y=x^2 + z^2$ with $0 \leq y \leq 1$, and the disk $x^2 + y^2 \leq 1$. Let $S$ have an outward orientation. Compute the double integral of $\langle ...
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votes
0answers
29 views

How does this integration by parts work: $\int_{Q}v\varphi_t\;dxdt = -\int_S \varphi v|_{S} \nu_t - \int_Q v_t \varphi\;dxdt$

Let $\Omega(t)$ be a bounded domain for each $t$. Let $Q=\bigcup_{t \in [0,T]} \Omega(t) \times \{t\}$ and $S=\bigcup_{t \in [0,T]} \partial\Omega(t) \times \{t\}$. The normal vector to $S$ at ...
0
votes
0answers
52 views

Surface Area cut from cone by cylinder

I genuinely have no idea how to do this. I've been struggling on this problem for about two hours now (more hours last night), and am not getting anywhere. The question is: Find the surface area ...
3
votes
1answer
41 views

Evaluating a surface integral of a paraboloid

Calculate the average value of $(1+4z)^{3}$ on the surface of the paraboloid $z=x^{2}+y^{2}$,$x^{2}+y^{2} \leq 1$ I'm not sure on how to start this problem. I have already found the area of the ...
0
votes
0answers
40 views

How to parametrise this surface integral

This is the question: $ S $ is the boundary of the region $ \{(x,y,z):0≤z≤h, a^2 ≤x^2+y^2 ≤b^2 \}$ where $ h,a,b$ are positive and $a<b$. ${\bf F(r) } = \exp(x^2+y^2){\bf r}$ where $ {\bf ...
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votes
1answer
27 views

Multivariable Calculus Surface Integral Calculation

I have a surface bounded by $x^2+y^2=1$ and $x^2+y^2=9$ (cylinders) as well as the planes z=0 and z=3.The vector field is $(yx^3,xy^3,x)$. I know this involves the divergence theorem, where I would ...
0
votes
2answers
34 views

Three Surface Integrals

Could someone assist with the following three surface integrals? Q1 The portion of the cone $z=\sqrt{x^2+y^2}$ that lies inside the cylinder $x^2+y^2 =2x$. Q2 The portion of the paraboloid ...
2
votes
2answers
465 views

Surface area of intersection of two cylinders

Let $$R=\{(x,y,z):y^2+z^2\leq 1\,\, \text{and}\,\, x^2+z^2\leq 1\}.$$ Compute the volume of $R$. Compute the area of its boundary $\partial R$. I'm fine with #1. For #2, I have a ...
0
votes
0answers
32 views

Integration by substitution on surface

Suppose I have an integral on the boundary of a Lipschitz domain $\Omega \subset \mathbb{R}^n$ $$\int_{\partial\Omega} f(x-y)dS$$. where $dS$ is the surface element $dS = g(x)dx$. Can I do a ...
1
vote
1answer
57 views

Bounding an integral on boundary of Lipschitz domain

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain with bounded boundary $\Gamma.$ So $\Gamma$ is a hypersurface of dimension $(n-1)$. I want to show that $$\int_\Gamma ...
1
vote
1answer
82 views

bounds for surface integral of a plane

I need to calculate the next surface integral, but i'm having troubles with the bounds; $$\iint -2 \, dS,$$ where $S$ is the part of the plane $x+2y+z=2$ that is cut off in the first octant. My ...
1
vote
0answers
106 views

Question about integral on hypersurface

Let $S$ be a (n-1) dimensional hypersurface in $\mathbb{R}^n$. I see something like this written: $$\int_S f(s)d\sigma(s)$$ where $d\sigma$ means the surface measure. Now if $\Phi\colon T \to ...
1
vote
2answers
29 views

Does $g(x,y,z)$ (the equation of the surface) need positive $z$ or negative $z$ when doing a surface integral?

$\quad$If a smooth surface $S$ is defined by $g(x,y,z)=0$, then recall that a unit normal is $$\mathbf{n}=\dfrac{1}{\|\nabla g\|}\nabla g,\tag{9}$$ where $\nabla g=\dfrac{\partial g}{\partial ...
1
vote
2answers
55 views

Surface Integral of a Vector Field Over a Torus

Let $S$ be the surface obtained after rotating $(x-2)^2+z^2=1$ around the $z$-axis. What is the value of $$\int_{S}\mathbf{F\cdot n } dA$$ where $$\mathbf{F}=(x+\sin(yz), y+e^{x+z}, z-x^2\cos(y))$$
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votes
1answer
36 views

Computing the surface integral of a parabloid

Problem: Solution: I am having difficulty understanding how the author determined the limits of integration of $R$. The author used $\theta=\pi/3\quad to\quad \theta=\pi/2$ and $r=1\quad to\quad ...
1
vote
1answer
46 views

Find an integral for the area of the surface generated by revolving the curve $y=sin(x)$ between $0 \le x \le \pi$, about the x-axis

So here is my problem: Find an integral for the area of the surface generated by revolving the curve $y=sin(x)$ between $0 \le x \le \pi$, about the x-axis Just thinking about the problem I feel ...
1
vote
0answers
109 views

Surface Integral

The glass dome of a futuristic greenhouse is shaped like the surface $z = 8 - 2x^{2} - 2y^{2}$. The greenhouse has a flat dirt floor at $z = 0$ Suppose that the temperature T, at points in and around ...
0
votes
1answer
63 views

Volume of hyperboloid limited by two planes (multivariable calculus).

As usual the teachers solution sheet takes leaps and bounds over steps in the solution that I need to understand it. Q: Determine the volume of the body limited by $x^2+y^2-3z^2=1$, $z=1$ and ...
0
votes
0answers
54 views

Did I solve this surface area of revolution problem correctly $y=\frac{1}{3}x^3,$ when $0 \le x \le 2$

My Professor hasn't posted the solutions to our practice exam just yet, but I'd like to know if I solved the following surface area of revolution problem correctly? $y=\frac{1}{3}x^3,$ when $0 \le x ...
3
votes
0answers
45 views

Find the surface integral of $f=|x|-|y|$ over the part of $z=1-\frac{x^2}{M}-\frac{y^2}{N}$ inside a cylinder.

(a) Find the surface integral of $f=|x|-|y|$ over the part of $z=1-\frac{x^2}{M}-\frac{y^2}{N}$ inside the region $\frac{x^2}{M^2}+\frac{y^2}{N^2}=1$ (b) Find the surface integral of $f=|xy|$ over ...
1
vote
0answers
48 views

Gauss and Stocks teory

Given $\phi\in C^1(R)$, and we define the curve and surface $\gamma=${$(x,y):y=\phi(x),0\le x\le 1$} $S=${$(x,y,z):z=\phi(\sqrt{x^2+y^2}),x^2+y^2\le 1$} a.I need to prove that $A(S)=2\pi\int_\gamma ...
0
votes
1answer
192 views

How to integrate over an arbitrarily positioned spherical cap in spherical coordinates

If you want to integrate over the SURFACE of a spherical cap that is positioned in the way it is on wikipedia, this is rather simple. since it has azimuthal symmetry you get a factor $2\pi$ and for ...
15
votes
2answers
224 views

Area of supercircles, or how to integrate $\int_0^1 \sqrt[n]{1-x^n}dx$?

Martin Gardner, somewhere in the book Mathematical Carnival; talks about superellipses and their application in city designs and other areas. Superellipses(thanks for the link anorton) are defined by ...
2
votes
2answers
150 views

Finding surface area - integral of $\sqrt{1+z^2}$

Sorry about this, this is more of a "am I going the right way" question, there's a surface it goes: $$x^2+y^2-z^2=1$$ Now this is nice because $x^2+y^2=r^2=1+z^2$ thus $r=\sqrt{1+z^2}$ (I want the ...
5
votes
1answer
99 views

Is this function bounded? Next question about integral $\int_{\partial M} \frac{1}{||y-x||} n_y \cdot \nabla_y \frac1{||y-x||} dS_y$.

Let $\partial M$ be $C^2$ closed surface in $\mathbb{R}^3$, $M$ is open. Show that $$ f(x) = \frac{\int_{\partial M} \left| \frac{1}{||y-x||} n_y \cdot \nabla_y \frac{1}{||y-x||} \right| ...
8
votes
1answer
113 views

Surface integral of $2x+y+2z=16$

Here's the question: Find the surface area of the part of the plane $2x+y+2z=16$ bounded by the surfaces $x=0$, $y=0$ and $x^2+y^2=64$. So, I know I have to parameterize the surface ...
5
votes
1answer
201 views

Limit $\lim_{x\rightarrow x_0, x\in M} \int_{\partial M} \frac{1}{||y-x||} n_y \cdot \nabla_y \frac{-1}{||y-x||} dS_y$

Ok I had a question I think I can almost answer it but I miss one step: Let $\partial M$ be a closed surface in $\mathbb{R}^3$, $x_0 \in \partial M$ than show this limit: ...
3
votes
1answer
36 views

Does every convex subset of $\mathbb{R}^n$ have normal almost everywhere?

Does every convex subset of $\mathbb{R}^n$ have normal almost everywhere? I want to know if I have any convex subset $M$ of $\mathbb{R}^n$ is it meaningful to talk about integral of second kind ...
0
votes
1answer
128 views

Surface Integral directly

Surface integral that has me stumped. Q: Calculate $\int \int_{S} F \cdot dA$ Where $F(x,y,z)= xi+yj+zk$ S is the boundary of the region $x^{2}+y^{2} \leq z \leq (2-x^{2}-y^{2})^{1/2}$ oriented so ...
2
votes
1answer
107 views

tricky surface integral

I am studying for my final and my prof gave us review questions but with no answers so I am lost with this question. If anyone can help I would really appreciate it. Question: Find the area of the ...
1
vote
3answers
396 views

Interpreting the Surface Integral over a Vector Field

I have seen the fact that in certain instances, the Surface Integral over a Vector Field gives the quantity of fluid flowing through the surface in unit time (as in here, or in any standard Vector ...
1
vote
1answer
126 views

Stokes theorem problem to find alpha and beta so that I is independent of the choice of S

I have a question that I got half through but can't finish it. If anyone could help I would appreciate it. Question: let C1 be the straight line from (-1,0,0) to (1,0,0) and C2 the semi circle ...
0
votes
1answer
183 views

Find area of a curvilinear triangle that includes hyperbolic functions

We were given this question in class and I tried to compute it and it looks to be pretty crazy. Can anyone take a look and let me know if I did it correctly? I would really appreciate it. ...
1
vote
2answers
762 views

Find area of a simple, smooth, closed curve lying in a plane

I was given this question in class and I assume it is a spin off of Green's theorem for finding the area of a closed curve $\lambda$ in 2D but expanded to 3D I believe. Anyways I am pretty confused ...
1
vote
0answers
106 views

Computing the surface area of a (piecewise) polynomial parametric surface

I'm wondering what kind of numerical integration (e.g. Gauss-Legendre quadrature) I should use to compute the surface area of a (piecewise) polynomial parametric surface. There are two cases. Case ...
1
vote
3answers
124 views

Calculate Area of Surface

I am trying to calculate the area of the surface $z = x^2 + y^2$, with $x^2 + y^2 \le 1$. By trying to do the surface integral in Cartesian coordinates, I arrive at the following: $\int_{-1}^{1}dx ...
0
votes
1answer
391 views

Help me understand a surface integral question?

The question is: Evaluate the surface integral: $$ \iint\limits_S \, x^2yz\ \mathrm{d} S $$ Where S is part of the plane z = 1 + 2x + 3y that lies above the rectangle [0,3] X [0,2] I literally just ...
1
vote
2answers
628 views

What is the difference between surface area and scalar surface integrals?

What is the difference between the surface area of a paremetrized surface and the scalar surface integral of a function in $\mathbb{R}^3$? Are they not the same thing?
3
votes
4answers
372 views

parametrization of surface element in surface integrals

I don't understand this How $ dS = \sqrt{ \left ( \partial g \over \partial x\right )^2 + \left ( \partial g \over \partial y\right )^2 + 1 } \; dA \; \; $ ?? Is $ dA = dx\times dy$??
1
vote
0answers
935 views

Analytical calculation of the total surface of overlapping spheres

Let's say I have two spheres whose center's coordinates (cartesian) are 0,0,0 d,0,0 and both have radius R. I want to analytically calculate the total surface ...