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. JasonRox

    Integrate Surface Integral w/ Jacobian Transformation - Help Needed

    Everything was going fine until I bumped into this... (b^2*c^2*Cos[x]^2*Sin[y]^4 + a^2*c^2*Sin[y]^4*Sin[x]^2)^(1/2) ...integrate that with respect to y, for the boundaries y=0..Pi. A Jacobian Transformation would be a good start, but I have no idea what functions I would use to simplify...
  2. S

    Evaluating Surface Integral of f=x over Semi Sphere

    I need to evaluate the surface integral of f=x over a semi sphere. I know how to evaluate surface integral of a semi sphere but what are my steps in this case. As I found from books I should double integrate f = x with semi sphere limits. The problem is that I don't know how to start and...
  3. M

    Surface Area of Sphere above xy-Plane & in Cylinder

    Hi! I don't know how to approach this problem. I need a little bit of help please. Here is the problem: Find the surface area of that portion of the sphere x^2 +y^2 + z^2 =a^2 that is above the xy-plane and within the cylinder x^2 + y^2 = b^2, 0 \leq b \leq a
  4. N

    Problem with Spherical Surface Integral

    A\; =\; 4\dot{r}\; +\; 3\dot{\theta }\; -\; 2\dot{\phi } Now the surface integral integral is: \int_{}^{}{\left( ?\times A \right)\; •\; da} (the ? mark is a del operator and the dot over a variable means a unit vector) ?\times A\; =\frac{\dot{r}}{r\sin \theta }\left[ \frac{\partial...
  5. N

    Calculating Surface and Volume Integrals on a Sphere: A Problem-Based Approach

    "Find the surface integral of r over a surface of a sphere of radius and center at the origin. Also find the volume integral of Gradient•R and compare your results" Do I just integrate r to get (1/2)r^2 and plug some limits in since the r-hats equal one?
  6. D

    Calculating Surface Integral with Stoke's Theorem | -2pi/5 Answer Explained

    Here is the question: Evaluate the surface integral ∫∫s (X^4 + Y^4 + Z^4) dσ, where dσ is the surface element and S = { (X,Y,Z) : X^2 + y^2 + Z^2 = 1} I know you have to take the square root of 1 + (dz/dx)^2 + (dz/dy)^2 dxdy. And I got -2X/2Z and -2Y/2Z, respectively. Then, I must...
  7. H

    Evaluate Surface Integral: Solve x^2 + z^2 = 9, x=0, y=0, z=0 and y=8

    Could someone take a look at this please? Thanks ===== Q. Evaluate Integral A.n dS for the following case: A=(6z, 2x+y, -x) and S is the entire surface of the region bounded by the cylinder x^2 + z^2 = 9, x=0, y=0, z=0 and y=8. ===== Using Gauss' (or Divergence) Theorem: Integral A.n...
  8. L

    Surface Integral of Two Surfaces

    Hello! This is a question from one of our past exams, and it's had me stumped for the past hour. The question states: The cylinder x^2+y^2=2x cuts out a portion of a surface S from the upper nappe of the cone x^2+y^2=z^2. Compute the surface integral: \int\int (x^4-y^4+y^2z^2-z^2x^2+1)...
  9. W

    What is the relationship between curl and surface integrals in vector calculus?

    Curl and Surface Integral (help!) Hello people! I've been working on this problem, but I can't find how differentials of V on the left side of the equation appear. *** Show, by expansion of the surface integral, that (see attached image). Hint: choose the volume to be a differential...
  10. K

    Need help on easy surface integral

    Yo guys, I'm stumped on how to parameterize this surface and then compute an integral over it I'm supposed to compute \int\int_S \vec{r} over the surface formed by the x-y plane and z=4-(x^2+y^2), but I don't know to put it together and do it (No matter how you work with x and y, z will...
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