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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 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 with $F=<z,y \sqrt{x^2+z^2},-x >$ through the surface given by : $D=(x^2+6x+z^2\le 0, -1\le y \le 0)$ So they should be a cylinder (height on $y$ axis) shifted by ...
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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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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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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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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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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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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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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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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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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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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 ...
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Surface integral and parametrization

I'm struggling with surface integrals, and I still do not have much confidence with the parameterization of functions. This is the exercise I would like to solve: Calculate the surface integral of ...
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Applications of Green's theorem

Let $f\in C^1$. Prove that for every $x_*$ : $(\nabla \times f)(x_*)=\lim_{\epsilon \rightarrow 0} \frac{1}{\pi \epsilon ^2} \int_{\partial B_\epsilon (x_*)}f(x)dx$ I know that $$ \lim_{\epsilon \...
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Integration over a surface are

When I start trying to solve this problem, I don't know where I go wrong. I think it should be in what I consider as my $r(\theta, \rho)$ $r(\theta,\rho)=(\rho cos(\theta), \rho sin(\theta), 0)$, $0\...
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Solve the following integral using Stokes Theorem.

I am asked to evaluate the following integral: $$\int\int \text{curl} \ \vec{F} \cdot d\vec{S}$$ where $F = (y,-x,zx^3y^2)$ and $S$ is the surface given by $x^2 + y^2 + 3z^2 = 1$ with $z \leq 0$. I ...
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Area of a Sphere using a Circle and Surface integral

When considering the surface $S: x^2+y^2+z^2 = R^2$ we know that the surface integral $$ \iint_S dS = 4\pi R^2$$ Since this is the area of a sphere, but while using surface integral I know that the $...
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How can we show that the limit of the following surface integral is finite?

I have an arbitrary parameterized surface $S(u,v)$ in Cartesian coordinates (with finite area) and the surface passes through origin. There is a very small circular hole of radius $\epsilon$ at the ...
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Surface Integral of Sphere between 2 Parallel Planes

A circular cylinder radius $r$ is circumscribed about a sphere of radius $r$ so that the cylinder is tangent to the sphere along the equator. Two planes each perpendicular to the axis of the cylinder, ...
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calculate $\iint z dS$ where S is the upper hemisphere of radius a.

I came across the following problem in my textbook and my answer differs from the one given and I just wanted to check my work to see where I went wrong calculate $\iint z dS$ where S is the upper ...
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Find the range of surface integral using spherical coordinates

Let $S$ be a section of sphere $x^2+y^2+z^2=3$ with $x\ge1$ and $y\ge1$(like wedge shape). Compute the area of $S$ finding the range of surface integral over $S$ via spherical coordinates. If the ...
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Finding an example path in a conservative vector field

If I know that in a conservative vector field has path independence, how would I go about finding an example path given the answer? What does F.ds represent and how does it equal to pi?
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Surface integral of the intersection of a cylinder and a surface

Let $\mathcal S$ the surface of cartesian equation $z=f(x,y)=x^2-y^2$ and $$V=\left\{(x,y,z)\in \mathbb R^3 : x^2+y^2<4\right\}$$ Write the parametric equation of the portion of surface obtained ...
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Verification of Gauss' Divergence Theorem visualisation

I'm having trouble visualising what the information provides specifically this part: Φ : [0, 1] × [0, 2π] → R^3 Does this mean the hemisphere has height of 1 in the z-axis and a radius of 2pi in the ...
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The surface integral $\int_S z^2 \, dS$ over the cube $S$

Evaluate the integral $$\int_S z^2 \, dS ,$$ where $S$ is the surface of the cube $\{-1 < x < 1, -1 < y< 1, -1< z< 1\}$. So I gather that it has six sides. So what I did was ...
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Parameterization of the portion of a cylinder between two planes

I have to parametrize the lateral section of the cylinder $\frac{x^2}{4} + \frac{y^2}{9}=1$ between the planes $z = 1-x$ and $z = 0$. I have $r(u,v) = (2\cos{uv},3\sin{uv},\frac{3v}{2} + \frac{3}{2})$...
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Evaluating the Surface Integral on a Sphere for a scalar function (integral is involved in PDE)

I will begin by saying that I don't want to dissuade anyone who doesn't know PDE from helping, so if you're just here for the integral, you can skip down to "Where I'm Stuck" (For future reference: $\...
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Surface integral finding the $u$ and $v$ vectors

While calculating a surface integral, how do you know which variable to use as $u$ and $v$ in the calculation? For example in Cartesian coordinates you have $x$, $y$ and $z$. In cylindrical $\theta$,...
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Solid Angle Limits

So I was trying to compute the are of a sphere. There are many ways to do this like integrating small rings on the sphere etc. I was looking to do this using the notion of a solid angle. What I know ...
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Verifying Stokes' Theorem for an upper hemisphere

There is a hemisphere, radius $1$, centred at $(0,0,0)$, where the vector field is $$\vec F = \Big(x^3+\frac{z^4}{4}\Big) \hat i + 4x \hat j + (xz^3+z^2) \hat k$$ Verify Stokes' theorem for this ...