Surface Integral Homework: A.n dS in Plane 2x+y=6, z=4

In summary, the problem is to find the integral of A.n dS, where A is (y,2x,-z) and S is the surface of the plane 2x+y=6 in the first octant cut off by the plane z=4. The approach to solve this problem is to project onto a plane and use the normal vector to find the integral. However, there is confusion about which plane to project onto. The question also asks for any ideas or help.
  • #1
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Homework Statement



So trying to find the Integral of A.n dS where A is (y,2x,-z) and S is the surface of the plane 2x+y = 6 in the first octant cut off by the plane z=4

Homework Equations





The Attempt at a Solution



So i always solve these by projection...but I am a bit confused this time..

normally the surface is in the form z=f(x,y) so i do z-f(x,y) and take grad to find the normal..

so is the normal vector here just (2,1)? ie. grad 2x-y-6 = 0?

In which case is the integral just the double integral of (y,2x,-4).(2,1,0) dA?

Im a bit confused..Any help would be great!
 
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  • #2
Any ideas..?
I'm pretty confused about which plane you project onto to solve this...

Thanks!
 
  • #3
So do you project onto y-z plane?

and is the integral therefore the 1/2 times the double integral of A.(2,1,0) dydz?
 

1. What is a surface integral?

A surface integral is a mathematical concept that calculates the flux, or flow, of a vector field across a surface. It represents the total amount of a vector quantity passing through a surface, similar to how a line integral represents the total amount of a scalar quantity along a path.

2. How do you calculate a surface integral?

To calculate a surface integral, you first need to define the surface and the vector field. Then, you use a specific formula, such as the flux integral formula, to integrate the vector field over the surface. This involves breaking the surface into small pieces, calculating the flux for each piece, and summing them up to get the total surface integral.

3. What is the significance of "A.n dS" in the given surface integral homework?

The notation "A.n dS" represents the dot product of the vector A and the unit normal vector n, multiplied by the differential surface area dS. This is used in the formula for calculating flux and helps to determine the direction and magnitude of the vector field passing through the surface.

4. Can you explain the given surface for the homework, 2x+y=6 and z=4?

The given surface is a plane defined by the equations 2x+y=6 and z=4. This means that the surface is a flat, two-dimensional shape that intersects the x, y, and z axes at specific points. The equation 2x+y=6 determines the shape of the surface in the x-y plane, while the equation z=4 determines the height of the surface in the z direction.

5. How can surface integrals be applied in real-world situations?

Surface integrals have many applications in physics, engineering, and other scientific fields. They can be used to calculate the flow of fluids, the amount of heat transfer across a surface, and the electric or magnetic flux through a surface. They are also useful in calculating surface areas and volumes of complex shapes, such as in 3D printing or computer graphics.

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