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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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Differential form of surface integral equation

Considering a scalar field in a plane (pressure vs. location) $P({\rm r})$ where ${\rm r}=(x,y)$ then the following surface integral gives the surface deformation due to the pressure in the elastic ...
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Equivalent form of a vector area of a surface

I am interested in showing that the vector area $$\int_{\mathcal{S}}da$$ can be equivalently given by $$\int_{\mathcal{S}} da = \frac{1}{2}\oint(r \times dI).$$ I am mostly interested in getting a ...
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Parameterizing a surface area in the first octant

So I stumbled across an exam question where it gives a surface area where: $$S = \{(x, y, z) : x, y, z ≥ 0, 2x + y + 2z = 4\}.$$ It then asks to sketch this surface area and we can see it's a line ...
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Compute the flux of the vector field $F(x,y,z) = \left(2x-y^2\right) \mathbf i +\left( 2x - 2yz\right) \mathbf j + z^2 \mathbf k$

Compute the flux of the vector field $F(x,y,z) = \left(2x-y^2\right) \mathbf i +\left( 2x - 2yz\right) \mathbf j + z^2 \mathbf k$ through the surface consisting of the side and bottom of the cylinder ...
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Surface integral of vector field over a quarter of a cylinder

This is a question set by my maths tutor, I answered it using the divergence theorem to get an answer of 18 pi, which is correct. But I was wondering how you would be able to get the same answer by ...
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Evaluate the Surface Integral $xyz$ $dS$ where $S$ is the surface defined by $2y=\sqrt{9-x}$, $x>0$, and between $z=0$ and $z=3$.

Evaluate the Surface Integral $xyz$ $dS$ where $S$ is the surface defined by $2y=\sqrt{9-x}$, $x>0$, and between $z=0$ and $z=3$. Don't even know where to start with this question.
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Surface integral over the surface of the cone $z=1-\sqrt{x^2+y^2}$ lying above the $xy$-plane and normal making an acute angle with $\vec k$

Let $\vec F=(x^2+y-4,3xy,2xz+z^2)$ and $S$ be the surface of the cone $z=1-\sqrt{x^2+y^2}$ lying above the $xy$-plane and $\vec n$ is the unit normal to $S$ making an acute angle with $\vec k$ , then ...
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How do I continue this surface integral?

I tried to solve this. After find normal vector, I don't know how know how do I continue this.. Question is in this pic. =>]1
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Area of parametric surface (theory)

In the picture below $\left \|\Delta u_i r_u \times \Delta v_i r_v \right \|$ is the area of the parallelogram $\Delta T_i$ Can someone please explain why the sides of the parallelogram $\Delta T_i$ ...
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Surface Integral

Integrate $f=\frac{Y}{X}\sqrt{4Z^2 + 1}$ over the portion of the paraboloid $Z=X^2+Y^2$ that lies above the rectangle with the following limits: $1\lt X\lt e , 0\lt y\lt 2$ in the $X-Y$ plane." I ...
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Finding the volume of 3 dimensional region under the graph of a function.

Im trying to do the following question but im confused. Let W be the three dimensional region under the graph of the function $f(x,y) = \mathrm{e}^{x^2+y^2}$ and over the region in the $(x,y)$ plane ...
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Evaluate the surface integral $\int \int z \cos \gamma \, dS$ over an unit sphere.

Evaluate $\iint z \cos \gamma \, dS$, over the outside of the $unit$ $sphere$ centred at origin, where γ is the inclination of the normal surface at any point of the $unit$ $sphere$ with the z-axis. ...
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Surface area of elliptic paraboloid

I'm trying to write the surface area of the part of the paraboloid: $$z=(1/2)x^2+(1/2)y^2$$ where $$z\leq a^2/2$$ as double integrals in Cartesian and Cylindrical coordinates. For Cartesian ...
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How to find the projected area in the x-z plane of an ellipsoidal cap rotated by angle β in x-y plane?

I have ellipsoidal cap rotated in the x-y plane by an angle $\beta$; where the axis size in x coordinate is 'a', the axis size in y-coordinate is 'b' and axis size in z coordinate is 'c'. I am trying ...
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Surface integrals: Finding the center of mass of a thin sheet with the shape of surface S

How do I find the center of mass of a thin sheet when $S$ is the upper hemisphere $x^2+y^2+z^2=a^2$ with $z\ge 0$ and density $\delta(x,y,z)= k$ (constant). Also I have to compute this using surface ...
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Why isn't the answer of this surface integral $24$?

Evaluate the surface integral $$\int_{S} (z + x^{2}y)\,dS.$$ $S$ is the part of the cylinder $y^{2} + z^{2} = 4$ that lies between the planes $x = 0$ and $x = 3$ in the first octant. I did the ...
If velocity vector is given as $\mathbf F=y\mathbf i +2 \mathbf j+\mathbf k$ , then find the flux of water through the parabolic cylinder $y=x^2$, $0\le x\le 3$, $0\le z \le 3$. For this problem I ...