3
votes
0answers
46 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 $k=0.2$): ...
-4
votes
2answers
37 views

Describe the surface whose equation is given by $x^2+y^2+z^2-y=0$. [closed]

Describe the surface whose equation is given by $x^2+y^2+z^2-y=0.$ Show your working.
1
vote
1answer
59 views

Is it true that $d\textbf{S} = dy dz\textbf{ i }+ dx dz\textbf{ j }+ dx dy\textbf{ k }$

I came up with this in my mind, Just wondering if it is true I am thinking about it too, will post my observations, if any
0
votes
1answer
44 views

Stokes' Theorem and Surfaces

Stokes' Theorem states the following: \begin{equation*} \oint_c \textbf{F}\centerdot d\textbf{r}= \int\int_S (\nabla \times\textbf{F})\centerdot nd \textbf{S}\end{equation*} for a given C that is the ...
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
2answers
291 views

Boundary under transformation of a closed curve from $R^2\to R^3$

Consider some mapping $\phi: R_{uv} \to S\subset \mathbb{R}^3$ where $R_{uv}\subset \mathbb{R}^2$ and such that it is a simply connected region. We call the boundary of the surface (which we ...
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 = ...
0
votes
1answer
36 views

Understanding surface integrals

My question is a bit vague, but I'm trying to get a better understanding of surface integrals and their relation to physics. Suppose I have a surface, say a sphere, and I have a function which gives ...
1
vote
1answer
42 views

Tangent Planes and Surfaces (Calc 3)

I am wondering if I am on the right track for the following question: Find a for the plane $x+y+z=-1$ so that it is a tangent plane to the surface $z=x^2+ay^2$ I figured since you are given a ...
2
votes
1answer
43 views

Surface Area Line integral problem

I'm trying to figure out how to solve a surface area with surface and line integrals (showing both methods). The area I'm trying to compute is the area of the shape $$x^2+y^2=9$$ bounded by $z=0$ and ...
0
votes
2answers
72 views

Surface Area of Two Cylinders Calculus 3

Find the surface area of two cylinders $$y^2 + z^2 = 1$$ and $$x^2 + y^2 = 1$$ I have so far set the two equations to equal $$x= \pm z$$ and $$y= \sqrt{(1-z^2)}$$ I am a little confused on how to set ...
0
votes
1answer
53 views

Find normal vector for the surface F(x,y)=0

I need to find the normal vector(in a point (a,b)) for a surface F(x,y)=0, that we can't write as y=f(x) and F(x,y) doesn't satisfies the conditions of the implicit function theorem. For example: the ...
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 ...
1
vote
2answers
86 views

How to parametrize this region surface

$S$ is the portion of the plane $$x+2y-3z=3$$ in the octan bounded by the positive direction of the $x$ and $y$ axis and the negative direction of the $z$ axis. How can I parametrize this crazy ...
1
vote
1answer
432 views

Verify Divergence Theorem (using Spherical Coordinates)

I'm very used to calculating the flux of a vector field in cartesian coordinates, but I'm still getting tripped up when it comes to spherical or cylindrical coordinates. I was given the vector field: ...
0
votes
2answers
86 views

parametrize surface region

S is the elliptic region of the plane $y+z=1$ inside the cylinder $4x^2+4(y-0.5)^2=1$. First parametrize $S$ using $(x,y,z)=G(u,v)$ and then calculate $\displaystyle \frac{dG}{du}\times ...
0
votes
2answers
62 views

parametrize a disc

$S$ is the disc of radius 1 centered at the origin located on the $xy$ axis, oriented downward. First parametrize the given surface using $(x,y,z)= G(u,v)$ with $(u,v)$ in $W$ and then calculate ...
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 ...
2
votes
1answer
93 views

Flux and Gauss theorem

I have a problem; There seems to be something wrong with my understanding of gauss theorem. Let's say $F = [y ; x^2y; y^2z]$. I want to calculate the flux of $F$ going out of $$D = \{1 \le z \le 2 - ...
2
votes
2answers
100 views

parametrization of plane in $\mathbb R^3$

Parametrize the plane in $\mathbb R^3$ with direction vectors $\hat u$ and $\hat v$ and through the point $p$ as in representation as the range of a $C^1$ function $f:\mathbb R^2\to\mathbb R^3$. ...
0
votes
1answer
305 views

Find the surface area obtained by rotating $y=1+3x^2$ from $x=0$ to $x=2$ about the y-axis.

Find the surface area obtained by rotating $y= 1+3 x^2$ from $x=0$ to $x = 2$ about the $y$-axis. Having trouble evaluating the integral: Solved for $x$: $x=0, y=1$ $x=2, y=13$ $$\int_1^{13} ...
1
vote
2answers
30 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 ...
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 ...
0
votes
1answer
45 views

Paremetric surface revolved around y-axis

if I'm finding the area of the surface generated by revolving the curve around the y-axis I use the equation $2\pi x\sqrt{(x')^2+(y')^2}$ and I'm given $$x=(2/3)t^{3/2}$$ $$y=2\sqrt{2}$$ and I got ...
1
vote
0answers
114 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
117 views

Recognize the equation of a surface of revolution

Yesterday, I asked a question about the critic points of the surface $$z = (x^2 + y^2)e^{-(x^2 + y^2)}$$ and my question was if I had a easier way to classify the critic points of this surfaces ...
1
vote
2answers
124 views

Maximum surface area of cylinder (1-variable)

In a given sphere of radius $R$, it is required to find the cylinder with maximum surface area that we can inscribe in this sphere. Using that the radius of the cylinder is $r$, with Pythagoras ...
0
votes
2answers
514 views

Projection of ellipsoid

Find the projections of the ellipsoid $$ x^2 + y^2 + z^2 -xy -1 = 0$$ on the cordinates plan I have no idea how to do this. I couldn't find much on google to help me with it too. Thanks in ...
1
vote
1answer
76 views

Why $xyz = e^x$ can be seen as the level surface $f(x,y,z) = xyz - e^x$?

That does not make sense to me. I recognize a level surface from the form $f(x,y,z) = k$. Where is the $k$ there? It looks just like a $3$ variables function to me.
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 ...
2
votes
1answer
150 views

Show that 2 surfaces are tangent in a given point

Show that the surfaces $ \Large\frac{x^2}{a^2} + \Large\frac{y^2}{b^2} = \Large\frac{z^2}{c^2}$ and $ x^2 + y^2+ \left(z - \Large\frac{b^2 + c^2}{c} \right)^2 = \Large\frac{b^2}{c^2} \small(b^2 + ...
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 ...
3
votes
0answers
83 views

Working with projection of areas?

I was recently solving a physics problem which had to do with the momentum imparted by a photon beam to a perfectly absorbing sphere and a perfectly reflecting one. Considering the former and Putting ...
1
vote
2answers
114 views

Find a point through which every surface tangent to z=xe^(y/x) passes

Find a point through which every plane tangent to the surface $$ z=xe^{\frac{y}{x}} $$ passes. It's not a homework. I know, that I need a normal vector and the point of tangency to find a tangent ...
2
votes
1answer
39 views

The connectivity of the intersection of hypersurface and ball

$u$ is a function defined on a connected open set $\Omega$ of $\mathbb R^n$ containing $0$ such that $u \in C^2(\Omega)$ and $u(0)=0$. Consider the hypersurface $X=\{(x,u(x))~|~x\in\Omega\}$ and the ...
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 ...
0
votes
2answers
1k views

Find surface area that lies above a triangle

Determine the area of the part of the surface $z=2 + 7x + 3y^2$ that lies above the triangle with vertices $(0,0)$, $(0,8)$, and $(14,8)$. I do not know what formula to use to attempt this problem!
1
vote
1answer
158 views

Compute the surface area of an oblate paraboloid

Consider the surface S: $z=4-4x^2-y^2, z\geq0$. Compute its surface area. I've tried the following: $Area(S)=\int\int_D \sqrt{(8x)^2+(2y)^2+1}dxdy$ with D being the interior of the ellipse ...
1
vote
0answers
37 views

Why does the surface area integral need the arc length differential but the volume doesn't? [duplicate]

When calculating the surface area of a revolution you need to use the arc length differential $$\sqrt{1 + y'^2}$$ but you don't need to use that when calculating the volume. Why is that? Thanks!
0
votes
1answer
38 views

Approximated distance between two points on a surface

I'm reading this paper and I don't understand this line because I haven't the book (I can't look up Theorem 7.4.2) . How come that the distance between $P_i$ and $P_{i-1}$ is calculated as follows?
1
vote
1answer
60 views

Area of a 3D surface

I need to compute the area of a $3D$ sphere centered on $0;0;0$ and the book I'm following says: "If a curve $y = f (x)$ from $y = a$ to $y = b$ is revolved around the $x$ axis, the surface area of ...
1
vote
2answers
138 views

Surface of revolution using cylindrical polars

Consider the surface of revolution of the curve $$y = x^2$$ where $0 < x < 1$. By writing a suitable integral, show that the area of this surface is 3.81 units. (You are advised to work in ...
0
votes
2answers
242 views

How to parameterize a hyperboloid in a solid of revolution

The middle “hyperboloid” part of the solid of revolution is determined entirely by a single edge of the cube that does not touch one of the axis vertices - there are six such edges. Mark these ...
4
votes
0answers
204 views

Understanding surface area of a revolution/length of curve

I don't quite understand why the formula to find the surface area of a revolution is what it is: $$A = 2\pi \int_a^b x\ \sqrt{1 + \left(\frac{\text{d}y}{\text{d}x}\right)^2}\ \text{d} x.$$ I ...
2
votes
2answers
2k views

How to find surface area of $x=\sqrt{a^2-y^2}$

I still hard time to find surface area of function... I have The given curve is rotated about the $y$-axis. Find the area of the resulting surface. $$x= \sqrt{a^2-y^2},\quad ...
1
vote
2answers
838 views

Surface area formula

I'm kind of confused about the explanation of the surface area formula in my text book The text gave us $$\int_{a}^{b}2\pi f(x) \sqrt{1+[f'(x)]^2}dx$$ after that the formula is getting like ...
1
vote
0answers
937 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 ...