What is Surface integral: Definition and 260 Discussions

In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate a scalar field (that is, a function of position which returns a scalar as a value) over the surface, or a vector field (that is, a function which returns a vector as value). If a region R is not flat, then it is called a surface as shown in the illustration.
Surface integrals have applications in physics, particularly with the theories of classical electromagnetism.

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  1. F

    Projecting a surface integral in 3-D onto a plane (2-D)

    Homework Statement So the context is this arises in the method of descent, for finding a solution for the 2D heat equation from the 3d heat equation. Anyway, in one step, we must change the surface integral over a ball in 3d, to the surface integral over it's projection into a plane. In this...
  2. X

    Evaluating a surface integral using parametric/explicit representation

    [EDIT]: Correct answer for this problem is 1/2, not 4 as I thought before; that means the result for the explicit representation was correct. Still I don't understand how to treat the case with the parametric representation. Greetings, I need to evaluate $$\iint_{S}\mathbf{F}\cdot\mathbf{n}\...
  3. U

    Not sure if this surface integral is right

    Homework Statement The Attempt at a Solution I did the manual integration of part (i) and got an answer of 5/6 instead, I'm not sure which part is wrong.. For the surfaces, I start off with the surface in the x-z plane, then the slanted plane, then the y-z plane, then the top of the prism...
  4. B

    Surface Integral Homework: Compute \iint_S \sin y dS

    Homework Statement Compute the integral \iint_S \sin y dS where S is part of the surface x^2 +z^2 = \cos^2(y) lying between the planes y=0 and y=\pi/2. Homework Equations \iint_S f(x,y,z) dS = \iint_D f(x,y, g(x,y)) \sqrt{g_x^2 +g_y^2 +1}dA \iint_S f(x,y,z) dS = \iint_D...
  5. M

    Surface integral and divergence theorem over a hemisphere

    Homework Statement Please evaluate the integral \oint d\vec{A}\cdot\vec{v}, where \vec{v} = 3\vec{r} and S is a hemisphere defined by |\vec{r}| \leqa and z ≥ 0, a) directly by surface integration. b) using the divergence theorem. Homework Equations -Divergence theorem in...
  6. D

    Surface Integral - 2 methods give different answer

    Thanks for checking this out. Here's the problem: I attempted to do it by using parametrize it into spherical coordinate. r(r,t) = (x= cost, y= sint, z=r) dS=|r_{u} x r_{v}| dA = r\sqrt{2} dA dA = rdrdt \int\intx^{2}z^{2}dS = \int\int\sqrt{2} cos^{2} r^{6} drdt I check my...
  7. M

    Why is z = rcos(θ) and not z = rsin(θ) in surface integrals?

    Look in the paint doc. I was wondering why they said z = rcos(θ) and not z = rsin(θ) and x = rcos(θ)
  8. countzander

    How Is the Surface Area of a Rotated Curve Calculated Using Integrals?

    Homework Statement Consider the surface S formed by rotating the graph of y = f(x) around the x-axis between x = a and x = b. Assume that f(x) ≥ 0 for a ≤ x ≤ b. Show that the surface area of S is 2π times integral of f(x)sqrt(1 + f ' (x)^2) dx from a to b. http://i.imgur.com/qFeGP.png...
  9. MacLaddy

    Surface Integral setup and evaluate

    Homework Statement Evaluate the surface integral ∫∫f(x,y,z)dS using an explicit representation of the surface. f(x,y,z) = x^2 + y^2;\mbox{ S is the paraboloid } z= x^2 + y^2\mbox{ for }0\leq z \leq 4 Homework Equations \displaystyle \int \int_{S} f(x,y,z)\ dS = \int \int_{D} f...
  10. MacLaddy

    Surface Integral: Find Area of Cone (0 ≤ z ≤ 4)

    Homework Statement Find the area of the following surface using an explicit description of a surface. The cone z^2=4x^2+4y^2\mbox{ for } 0 \leq z \leq 4 Homework Equations \iint_s f(x,y,z)dS=\iint_R f(x,y,g(x,y))\sqrt{z^2_x+z^2_y +1} The Attempt at a Solution I have solved the...
  11. S

    Surface Integral of a Cylinder

    Homework Statement Let S denote the closed cylinder with bottom given by z=0, top given by z=4, and lateral surface given by the equation x^2 + y^2 = 9. Orient S with outward normals. Determine the indicated scalar and vector surface integral to ∫∫ x^2 i dS (I have tried to solve this...
  12. M

    MHB Another surface integral: Evaluating a Surface Integral on a Paraboloid

    Here is another that I am stuck on. Please doublecheck my work, and let me know if where I am stuck is correct, or if I am on the completely wrong path. Evaluate the surface integral \(\int\int f(x,y,z)dS\) using an explicit representation of the surface. \(f(x,y,z) = x^2 + y^2;\mbox{ S is...
  13. M

    MHB Area of Cone \(z^2 = 4x^2 + 4y^2\) 0-4: Solve & Find Answer

    Hello all, I've hit a roadblock on a question regarding Surface Integrals. I seem to be having a problem conceptualizing many of these concepts. Anyway, here goes. Find the area of the following surface using an explicit description of a surface. The cone \(z^2 = 4x^2 + 4y^2\) for \(0\leq...
  14. Y

    Surface Integral over a Cone - Stokes?

    Homework Statement Homework Equations I'm guessing Stoke's Theorem? However, I'm not sure how to apply it exactly.. The Attempt at a Solution Looking at Stoke's Theorem, I'm still not sure how to apply it. I'm really just lost as to where to begin; is there even a \grad F to take? I know...
  15. C

    Surface integral (Flux) with cylinder and plane intersections

    Homework Statement Evaluate the surface integral \int_{S} \int \vec{F} \cdot \vec{n}\,dSwith the vector field \vec{F⃗}=zx\vec{i}+xy\vec{j}+yz\vec{k} . S is the closed surface composed of a portion of the cylinder x^2 + y^2 = R^2 that lies in the first octant, and portions of the planes...
  16. T

    Parameterizing surface in surface integral problem

    Homework Statement Find the area of the surface cut from the paraboloid z=2x2+2y2 by the planes z=2 and z=8.Homework Equations Surface area of S= ∫∫ ||Ts×Tt|| ds dtThe Attempt at a Solution What I am really having trouble doing in this problem (and in general) is parameterizing the surface in...
  17. T

    How a vol. integral becomes a vol. integral plus surface integral

    Hello, Please see this pdf at some universities website: http://physics.ucsc.edu/~peter/110A/helmholtz.pdf In line 14 the author claims using integration by parts...I do not understand who could the integration by parts be used here. I understand the general case where we have...
  18. O

    Divergence Integral doesn't equal surface integral

    We were given an electric field defined by Kr^3 , and asked to calculate what the total flux would be given a sphere of a radius R. I had already calculated the divergence of E to be equal to 5kr^2 . So the first integral is calculating what the divergence over the area of the sphere is...
  19. U

    Evaluate this surface Integral

    Homework Statement The problem is attached in the picture.The Attempt at a Solution The suggested solution went straight into the hardcore integration. I was trying a different approach by changing the variables (x,y) into (u,v) which appear to make the integration much easier... The...
  20. R

    Evaluate surface integral over surface

    Homework Statement Evaluate the surface integral of G over the surface S S is the parabolic cylinder y=2x^2, 0=< x =<5, 0=< z =<5 G(x,y,z)=6x Answer is one of the following: 1. (15/8)*(401sqrt(401)-1) 2. (5/8)*(401sqrt(401)-1) 3. (15/8)*(401sqrt(401)+1) 4. (5/8)*(401sqrt(401)+1)...
  21. S

    Surface Integral finding the limits

    Homework Statement I need to evaulate ∫ ∫S dS where S is the surface z = x² + y², 0 ≤ z ≤ 4. Homework Equations dS = √( 1 + ƒ²x + ƒ²y)dxdyThe Attempt at a Solution dS = √( 1 + 4x² + 4y²)dxdy here's the problem what are the limits to the surface integral? no clue.. dx means i should find...
  22. R

    Relationship between two surface integral equations

    Homework Statement Calculate the surface integral ∫∫S x2z2 dS, where S is the part of the cone z2=x2+y2 between the planes z=1 and z=3 Homework Equations There are two relative equations for calculating surface integrals by transforming them into double integrals, but my question is about...
  23. Y

    Computing a surface integral with polar coordinates

    Homework Statement Show that ##\iint_{S}(x^2 + y^2)d\sigma = \frac{9\pi}{4}## where ##S = \{(x,y,z): x > 0, y > 0, 3 > z > 0, z^2 = 3(x^2 + y^2)\}## Homework Equations ##\iint_{S}f(x,y,z)d\sigma = \iint_{R}f(r(x,y))\sqrt{[r_x(x,y)]^2 + [r_y(x,y)]^2 + 1}## where ##r : R → ℝ^3, R \in ℝ^2##...
  24. ElijahRockers

    Surface Integral of a helicoid.

    Homework Statement Evaluate \int\int_{S}\sqrt{1+x^2+y^2} dS S is the helicoid with vector equation r(u,v) = <u cos(v), u sin(v), v> 0<u<2, 0<v<4pi The Attempt at a Solution If I replace the term under the radical with its vector equation counterpart, and multiply that by the...
  25. L

    Where have I gone wrong with this surface integral problem?

    Homework Statement Evaluate the surface integral ∫F.dS where F = xi - yj + zk and where the surface S is of the cylinder defined by x^2+y^2≤4, and 0≤z≤1. Verify your answer using the Divergence Theorem. Homework Equations The Attempt at a Solution I parametrized the surface in...
  26. L

    Surface Integral, flux. Boundary and orientation

    In solving a flux integral over a flat surface, inclined above the xy-plane, does the boundary of the surface influence the flux only through the integral limits? (and not through its normal vector) Let's say that there is an elliptic surface inclined above the xy-plane. The orientation is...
  27. H

    Surface integral problem from H.M. Schey's book

    I've been fooling around by myself with the book "div, grad, curl and all that" by H.M. Schey to learn some vector calculus. However, in the second chapter, when he performs the integrals, he skips the part where he finds the limits on x and y. Here's an example: Compute the surface integral...
  28. K

    Surface integral to Lateral integral?

    Homework Statement I have some working out my lecture gave me to a problem and I don't think I understand part of it. Hoping you could help me. It's using Gauss' Law to find the capacitance of a cylindrical capacitor of length L but this information shouldn't matter for my question...
  29. A

    Surface integral or Divergence Theorem confused?

    Homework Statement Find the Volume ∫∫ xy DA where R is the region bounded by by the line y=x-1 and the parabola y^2=2x+6. Homework Equations ∫∫ xy dx dy The Attempt at a Solution first i found the intersection of the above equations . which is (5,4) to (-1,-2) . then i...
  30. A

    Surface Integral: Calculating Over a Square XY-Plane

    Homework Statement Calculate the surface integral of the vector field a=xy i + (x+1) j + xz^2 k over a square in the xy-plane with length 1 and whose unit normal points in the positive direction of the z axis. Homework Equations This is the problem. There are many different types of...
  31. ArcanaNoir

    Surface integral use stokes/divergence/whatever is convenient

    Homework Statement Consider the closed surface S consisting of the graph z=1-x^2-y^2 with z \ge 0 and also the unit disc in the xy plane. Give this surface an outer normal. Compute: \int \int_s \mathbf{F} \cdot d \mathbf{S} Homework Equations Stokes theorem, divergence theorem...
  32. K

    Time-retarded E-field and Gaussian surface integral

    If I have an oscillating charge inside of a sphere, will the integral of E(t), where t=proper time of the sphere, over the sphere's surface area result in a value of electric flux equal to the value of the charge?
  33. H

    Surface integral of a cylinder

    Homework Statement Evaluate the surface integral ∫A\bullet\hat{n}dS where A = z\hat{x}+x\hat{y}+3y^2z\hat{z} and S is the cylinder x^2+y^2=16 for the range of x\geq0,y\geq0, 0\leqz\leq5 Homework Equations I used this page as an example way to do this. I'm not good with surface integrals...
  34. DryRun

    Surface Integral of an Inverted Paraboloid with Radius 2 and Height 4

    Homework Statement http://s1.ipicture.ru/uploads/20120118/EHTTIkiQ.jpg The attempt at a solution I've drawn the graph in my copybook. It's an inverted paraboloid with radius 2 on the xy-plane height of 4 on the z-axis, which cuts off at plane z=3. ∇\vec{\phi}=-2x\vec{i}-2y\vec{j}-\vec{k}...
  35. B

    Evaluating a Surface Integral: xze^y i -xze^y j +z k

    Homework Statement Evalute the surface integral Homework Equations F(x,y,z)=xze^y i -xze^y j +z k for the surface is partof the plane x+y+2z=2 in the first octant and orientated downwards The Attempt at a Solution \displaystyle \int \int_{\sigma} F dS=\int \int_R (xze^y i -xze^y...
  36. B

    Evaluating Surface Integral for Hyperboloid in Cylinder

    Homework Statement Evaluate the surface integral Homework Equations f(x,y,z)=y where sigma is part of the hyperboloid y=x^2+z^2 that lies inside cylinder x^2+z^2=4 The Attempt at a Solution For \displaystyle S= \int \int_R \sqrt(z_x^2+z_y^2+1) dA I calculate...
  37. W

    Surface integral of a vector field w/o div. theorem

    Homework Statement First a thanks for the existence of this site, i find it quite useful but had no need to actually post till now. I am stuck on the following problem in "introduction to physics" We should calculate the \oint \vec{v}.d\vec{A} of a object with the following parameters...
  38. B

    Evaluating a Surface Integral: A Parallelogram

    Homework Statement Evlute the surface integral Homework Equations f(x,y,z)=x+y+z where sigma is the parallelogram with parametric equations x=u+v, y=u-v and z=1+2u+v where 0 <=u<=2 and 0<=v<=1. The Attempt at a Solution I have no idea how to tackle this. Any suggestions?
  39. B

    Evaluating Surface Integral for f(x,y,z)=(x^2+y^2)zy

    Homework Statement Evaluate the surface integral \displaystyle \int \int_\sigma f(x,y,z) dS for f(x,y,z)=(x^2+y^2)zy where σ is the portion of the sphere x^2+y^2+z^2=4 and abov plane z=1 The Attempt at a Solution I realize this can be done by parameterising the surface using θ and ∅...
  40. J

    Converting cartesian surface integral to polar

    If I have an integral: \int\int_{R} x^{2} + y^{2} dy dx Where the region R is the area enclosed by a circle centered on the origin of any given radius, is it possible to just convert x^2 + y^2 to r^2 and integrate from 0 to r over dr and 0 to 2 pi over d\theta? So it would become...
  41. I

    Surface integral problem - don't need to use Jacobian for polar?

    Homework Statement Evaluate the surface integral. ∫∫S x^2*z^2 dS S is the part of the cone z^2 = x^2 + y^2 that lies between the planes z = 1 and z = 3. Homework Equations \int \int _{S}F dS = \int \int _D F(r(u,v))|r_u\times r_v|dA x=rcos(\theta) y=rsin(\theta) The Attempt...
  42. D

    Surface Integral of Vector fields

    Homework Statement Use Stokes' Theorem to evaluate ∫C F · dr. C is oriented counterclockwise as viewed from above. F(x, y, z) = (x + y^2) i + (y + z^2) j + (z + x^2) k C is the triangle with vertices (9, 0, 0), (0, 9, 0), and (0, 0, 9). Homework Equations Stokes' Theorem The...
  43. I

    Surface Integral of a Cylindrical Surface

    Homework Statement What is the integral of the function x^2z taken over the entire surface of a right circular cylinder of height h which stands on the circle x^2 + y^2 = a^2 Homework Equations The Attempt at a Solution My problem is writing the equation in cylindrical form if...
  44. X

    What is the surface integral of a cone in polar coordinates?

    Homework Statement g(x,y,z) = z2; Ʃ is the part of the cone z = \sqrt{x2+y2} between the planes z = 1 and z = 3. Homework Equations Conversion to polar coordinates ∫∫Ʃg(x,y,z)dS = ∫∫Rg(x,y,f(x,y)) \sqrt{fx2 + fy2+1} The Attempt at a Solution If we're talking in terms of r and θ...
  45. H

    Surface Integral between planes

    Homework Statement ∫∫s x √(y2 + 4) where S: y2 + 4z = 16, and portion cut by planes x=0, x=1, z=0. Homework Equations I attempted to solve using the surface area integral formula, whereby this double integral is transformed to ∫∫f(x,y,g(x,y)) √((∂z/∂x)2 + (∂z/∂y)2 + 1) dA The...
  46. beyondlight

    Surface Integral Calculation for Parabolic Cylinder on Circular Area

    How do i calculate the surface integral ∫∫xdS where z=x^{2} is the parabolic cylinder over the area x^{2}+y^{2}=1I do not know how to solve this task because i can't express the surface parametrization in r(x,y). But when i express it as g(x,z)=0 i get the double integral dependet on dxdz...
  47. N

    Variable conversion in surface integral

    Hello helpful fellas, I'm reading an ecological model that involves setting up boundaries conditions. As part of a longer derivation, there is this flux equation (seen in attachment). Since I haven't formally studied vector calculus, please educate me in how it works and preferably with...
  48. M

    Surface Integral: dot product of two unit vectors

    Hello. :smile: I understand most of the work involved with these types of questions, but there is one point in an example I'm following that I don't understand. Homework Statement Evaluate: I = \int{(z^2)}dS over the positive quadrant of a sphere, where (x,y > 0). Homework...
  49. jegues

    Electric field vector & surface integral

    Homework Statement See figure attached. Homework Equations The Attempt at a Solution See figure attached. The solution shows that, \vec{r} = x \hat{i} + y \hat{j} + z \hat{k} How did they obtain this?
  50. D

    Surface integral of normal vector

    Hi. Does anyone know how to prove that \int \int dS \hat \mathbf n = \int \mathbf r \times d\mathbf r i.e., the surface integral of the unit normal vector equals the line integral on the r.h.s. ?
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