3
votes
0answers
53 views

Wicked domain of integration in a triple integral

I am dealing with a domain of integration of the form: $\left(\frac{x-y}{x+y}\right)^2+\left(\frac{y-z}{y+z}\right)^2+\left(\frac{x-z}{x+z}\right)^2\leq k$ The region looks like this (for $k=0.2$): ...
2
votes
1answer
56 views

Surface Integral over a sphere

Suppose $f(x,y,z)=g\left(\sqrt{x^2+y^2+z^2}\right)$, where $g$ is a function of one variable such that $g(2)=-5$. Evaluate $$\iint_S f ~dS,$$where $S$ is the sphere $x^2+y^2+z^2=4$. Now, I ...
2
votes
1answer
62 views

What is the meaning of $d\vec S$ in a surface integral?

Can someone explain if I have a surface $z= 9-x^2-y^2$ What would $\vec{n}$ be? What would $d\vec{S}$ be? Why is $d\vec{S}$ $(2x,2y,1)$ and not $(2x,2y,1)/\sqrt{4x^2+4y^2+1}$? Thanks!
1
vote
2answers
70 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
77 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
93 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
52 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
28 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
vote
2answers
31 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
16 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
12 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 $S = \{(x,y,z): x^2 +y^2 = 1, 0 \leq z \leq 1/2 \}$ ...
0
votes
1answer
23 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 ...
0
votes
0answers
46 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 ...
0
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 ...
4
votes
1answer
43 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
42 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 ...
0
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
533 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
58 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
85 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
110 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
31 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
56 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 $$\mathbf{F}=(x+\sin(yz), y+e^{x+z}, z-x^2\cos(y))$$
0
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
50 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
115 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
71 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
56 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
47 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
49 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
211 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 ...
14
votes
2answers
225 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
157 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
100 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
115 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
202 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
131 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
109 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
427 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
128 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
200 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
841 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
126 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
409 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
656 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?