Surface Integral Homework: ∫∫σ3x2 + 3y2 + 3z2 dS

In summary, the problem involves evaluating a surface integral over the part of a cylinder between the planes z = 0 and z = 1, including the top and bottom disks. The equation of the surface between the disks is a constant, and the integration can be done in cylindrical coordinates. The integral will need to be split into three separate parts to account for the sides of the cylinder.
  • #1
Baumer8993
46
0

Homework Statement



Evaluate ∫∫σ3x2 + 3y2 + 3z2 dS
where σ is the part of the cylinder x2 + y2 = 4 between the planes z = 0
, and z = 1, together with the top, and bottom disks.

Homework Equations


Surface integrals, maybe divergence theorem?


The Attempt at a Solution


I am having trouble knowing where to start with this one. I think I need to do a surface integral, but maybe with more than one surface? If that is right then what would I do for the cylinder side? How would I handle the z in the integral?
 
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  • #2
What is the value of z at the top and bottom of the cylinder?

What is the equation of the surface in between the top and bottom disks? Hint: it's a constant
 
  • #3
I would start the integration over the curved surface by converting to cylindrical coordinates.
 
  • #4
Ok so I see that z = 0, and z = 1. What about the sides? Do I have to do them in three separate integrals?
 
  • #5
Baumer8993 said:
Ok so I see that z = 0, and z = 1. What about the sides? Do I have to do them in three separate integrals?
Yes.
 

FAQ: Surface Integral Homework: ∫∫σ3x2 + 3y2 + 3z2 dS

What is a surface integral?

A surface integral is a mathematical concept used in multivariable calculus to integrate a function over a two-dimensional surface in three-dimensional space.

How do you solve a surface integral?

To solve a surface integral, you first need to parameterize the surface in terms of two variables, usually u and v. Then, you evaluate the function at each point on the surface and multiply it by the surface's differential area element, dS. Finally, you integrate this expression over the bounds of u and v to get the final result.

What is the significance of the function being integrated in a surface integral?

The function being integrated represents a physical quantity or property that is distributed over the surface. By integrating this function, we can calculate the total value of that quantity over the entire surface.

What is the difference between a surface integral and a double integral?

A surface integral is a type of double integral, but it is specifically used for integrating over a two-dimensional surface. A regular double integral, on the other hand, is used to integrate over a two-dimensional region in the xy-plane.

What are some real-world applications of surface integrals?

Surface integrals have many real-world applications, including calculating the flux of a vector field through a surface, determining the mass or charge of a three-dimensional object, and finding the center of mass of an object with variable density. They are also used in physics, engineering, and computer graphics to model and analyze various systems and phenomena.

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