How do you find the average value of a function on a surface?

In summary, the average value of a function on a surface is found by dividing the sum of all values of the function over the surface by the total surface area. The formula for this is (integral of function over surface) / (surface area). The average value can be negative if the majority of the function values are below the surface. It is equivalent to the average value of the function on a line if the surface is flat, but may differ if the surface is curved. Additionally, the average value can change if the surface is rotated or translated due to changes in surface area and function values.
  • #1
MeMoses
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Homework Statement


find the average value of (1+4z)**3 on the surface ...


Homework Equations





The Attempt at a Solution


All I really want to know is how to go about this? Is the average value on the surface just the surface integral over the area of (1+4z)**3?
 
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  • #2
It's the integral of (1+4z)^3 over the surface divided by the area of the surface.
 
  • #3
thank you, that makes more sense
 

Related to How do you find the average value of a function on a surface?

1. How do you define the average value of a function on a surface?

The average value of a function on a surface is the sum of all the values of the function over the surface, divided by the total surface area. This value represents the "center of mass" of the function on the surface.

2. What is the formula for finding the average value of a function on a surface?

The formula for finding the average value of a function on a surface is:
average value = (integral of function over surface) / (surface area)

3. Can the average value of a function on a surface be negative?

Yes, the average value of a function on a surface can be negative. This means that the values of the function are mostly below the surface and the center of mass is below the surface.

4. How is the average value of a function on a surface related to the average value of the function on a line?

The average value of a function on a surface is the same as the average value of the function on a line if the surface is a flat plane. However, if the surface is curved, the average value on a line may be different from the average value on the surface.

5. Can the average value of a function on a surface change if the surface is rotated or translated?

Yes, the average value of a function on a surface can change if the surface is rotated or translated. This is because the surface area and the values of the function over the surface will change, resulting in a different average value.

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