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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.

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. Given a surface, one may integrate over its scalar fields (that is, functions which return scalars as values), and vector fields (that is, functions which return vectors as values).

Read more on wikipedia's entry Surface integral.

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Use symmetry to solve surface integrals

Let S be an octant of sphere of radius $2$ centered at the origin. Precisely we take the surface $z=f(x,y)=\sqrt{4-x^2-y^2}$ and restrict to $x\geq 0$ and $y\geq 0$ we want to compute the surface integral over S of the function $z^2$. To solve it we…
Farouk Baly
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F in $\int_R {F ds}$ in surface integral

In high school in integral course I was told that in order to calculate the area of R in $\int_R {F ds}$ just put $F=1$ and integrate . I don't understand because $\int_R {F ds}$ ( if $R$ is the area of integration and $ds$ is an element of area )…
Alireza Amooei
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Surface area of a quadratic surface patch

I'm wondering about the surface area of the graph of $f(x,y)=xy$ for $x,y$ in an axis-aligned rectangle. The surface area is given by the integral $$ \int_m^n \int_p^q \sqrt{x^2 + y^2 + 1} \, dx \, dy $$ but this doesn't seem to simplify. Can…
Zach Teitler
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Parameterization of cone for surface integral

In the following parameterization of a cone: $$r(u,\theta) = u \cos v i + u \sin \theta j + u k, 0 \le u \le r$$ for a constant $r$, my understanding is that $\theta$ represents the angle that "goes around" the circle at a height $u$ from the…
Euclid
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Find bounds for problem about surface integral over triangle

Let $S$ denote the plane surface whose boundary is the triangle with vertices $(1,0,0), (0, 1, 0), (0, 0, 1),$ and let $F(x, y, z) = xi + yj + zk$. Let $n$ be the unit normal to S where $z \ge 0$. Evaluate the integral $\iint_S F \cdot dS$ using…
Euclid
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Find the surface area of the portion of $z^2=2xy$ where $0 \le x \le 2$ and $0 \le y \le 1.$

Find the surface area of the portion of $z^2=2xy$ where $0 \le x \le 2$ and $0 \le y \le 1.$ I tried parametrizing the surface as $$r(u,v) = ui+vj+\sqrt{2uv}k.$$ Using this, I got that $\lVert \frac{\partial r} {\partial u} \times \frac{\partial r}…
Euclid
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Compute the area of the region cut from a plane by a cylinder

Compute the area of the region cut from the plane $x+y+z = a$ by the cylinder $x^2 +y^2 = a^2$. The solution I am reading is here. I understand how they parametrized $x$ and $y$ in $r(u,v)$, but why is the parameterization of $z$ equal to $a−u \cos…
Euclid
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Problem with calculating a surface integral.

I try to calculate this surface integral, however my method does not work. Problem Attempted sol: I tried to project the surface at the $xy$ plane using this formula and the relationship $z = \sqrt{x^2-y^2}$. Projection of a surface However, then I…
Curtis
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Surface integral - undefined vector field

Consider the vector field $$\mathbf F(x,y, z) = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{[x^2+y^2+z^2]^{3/2}}$$ Let $S_1$ be the sphere given by $x^2 + (y-2)^2 + z^2 = 9$ oriented outwards. Compute $$\iint_{S_1}\mathbf{F}\mathbf{\cdot} \hat{\mathbf n}\…
Gummy bears
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Flux integral - parabloid inside a cylinder

Compute the flux integral where $F =<−0.5x^3 −xy^2 , −0.5y^3 , z^2>$ and S is the part of the paraboloid $z = 5−x^2 −y^2$ lying inside the cylinder $x^2 + y^2 \leq 4$, with orientation pointing downwards. After all the simplifications, the flux…
Gummy bears
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Calculate the surface integral given

Calculate the surface integral $ \ \large \int_{D} xyz dS \ $, where the surface $D$ is that part of the sphere $x^2+y^2+z^2=4$, which located above the area $y \leq x, \ y \leq 0, \ 0 \leq x^2+y^2 \leq 4$. Answer: $x^2+y^2+z^2=4, \ 0 \leq x^2+y^2…
MAS
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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 the function $$f = (x-1)^2 + (y-2)^2$$ extended to…
user3204810
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Surface comprising the union of two smooth surfaces?

I have a question at the moment which states that S is a solid spherical shell bounded by the surface H, comprising the union of two surfaces (which are spheres) which are: $x^{2}+y^{2}+z^{2}=1$ $x^{2}+y^{2}+z^{2}=0.5$ What on earth is the surface…
Sphero
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Finding surface area cut from a sphere

I wish to find the surface area cut from the sphere $x^2+y^2+z^2=2$ and the cylinder $x^2+y^2=1$ Here is what i tried: by uniting the 2 equations we get: $z=1$ The shape looks like a dome, when: $0\lt z\lt 1$ $0\lt r\lt \sqrt 2$ $0\lt \theta \lt…
segevp
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Surface integrals on Vector field

is it posibile to find the region of integration using $∫∫F.n dr$ if the surface projected on $xy-plane$ extends with out limits like the surface $z=x-y$. what i meant is if we project ,for example,$z=x-y$ it would be x=y whose variable does not…
serra
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