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

In summary, the integral is equal to 18pi when using Gauss' or Divergence Theorem for the given case of A=(6z, 2x+y, -x) and S being the entire surface of the region bounded by the cylinder x^2 + z^2 = 9, x=0, y=0, z=0 and y=8. The factor of 1/4 comes from the fact that the cylinder is cut into quarters by the planes x=0 and z=0, resulting in a quarter of the volume being considered in the integration.
  • #1
Hoofbeat
48
0
Could someone take a look at this please? Thanks

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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.
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Using Gauss' (or Divergence) Theorem:

Integral A.n dS = Integral Div A dV

Div A = 1, therefore

Integral 1 dV

When I last did this question, I said that dV = pi*r^2*h where r^2=9 and h=8. However, I've also got a factor of 1/4 to give me the final answer of 18pi. Where the hell did the quarter come from?! I'm guessing it has something to do with the limits and the fact that it's bounded by x=0, y=0 and z=0, but I don't understand why! Anyone offer some advice?!
 
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  • #2
Hoofbeat said:
Where the hell did the quarter come from?! I'm guessing it has something to do with the limits and the fact that it's bounded by x=0, y=0 and z=0, but I don't understand why! Anyone offer some advice?!

You think it right. The bases of the cylinder are parallel to the (xy) plane at y =0 and y=8. The (z,y) plane at x=0 cuts the cylinder into half. The other plane, (x,y) at z=0 does the same, so you have a quarter of a cylinder of length h=8.

ehild
 
Last edited:
  • #3
thanks ever so much! :)
 

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

What is a surface integral?

A surface integral is a mathematical concept used in multivariable calculus to calculate the flux of a vector field across a surface. It involves integrating a function over a two-dimensional surface in three-dimensional space.

What is the surface in this problem?

The surface in this problem is the circular cylinder defined by the equation x^2 + z^2 = 9. This surface lies in the x-z plane and has a radius of 3 units.

How do I evaluate a surface integral?

To evaluate a surface integral, you need to first parameterize the surface in terms of two variables (usually u and v). Then, you need to calculate the cross product of the partial derivatives of the parameterization. Finally, you integrate the given function over the parameterized surface using the cross product as the integrand.

What are the boundary conditions for this problem?

The boundary conditions for this problem are x=0, y=0, z=0 and y=8. This means that the surface integral is being evaluated over the portion of the cylinder that lies between the x-z plane and the plane y=8.

What is the physical significance of a surface integral?

A surface integral has various physical interpretations, depending on the context in which it is used. In physics, it can represent the flow of a fluid across a surface, in electromagnetism it can represent the electric flux through a surface, and in heat transfer it can represent the amount of heat exchanged across a surface. It is a useful tool for solving many real-world problems in science and engineering.

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