# 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 evaluate this integral over a slice of the unit disk?

I have a function $f(r)$ in polar coordinates, for some positive $a$ and $b$, with $0<n<1$, defined on the unit disk ($0<r<1$) $$f(r) = a + \dfrac{b}{(1-r)^n}$$ I would like to integrate ...
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### Surface area calculation of sphere segment.

I got this problem from a journal and curious how they have calculated that.Previously I asked this problem and unfortunately did not get any answer. Hereby I am posting again and hopefully someone ...
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### lateral surface area of cylinder

Use cylindrical coordinates and multivariable calculus to prove that the lateral surface area of a right, circular cylinder with radius 2 and height h is 4pih. I parameterized x = rcostheta, y = ...
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### parametrize the boundary of a region

I need to parametrize the boundary of this region : $D=\{y^2+z^2\le x^2+18,x^2+y^2\le 16\}$ So It's a one-sheet hyperboloid (radius=$\sqrt{18}$)+ cylinder with radius 4 I know how to parametrize ...
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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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### 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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### 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 $0 \leq u \leq \dfrac{1}{2}\tan v$ ...
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 ...