Averaging Polynomial & Non-Linear Functions: Examples & Explanations

AI Thread Summary
The discussion explores the concept of averaging in polynomial and non-linear functions, using two examples involving velocity and force to illustrate how average values can be calculated. It highlights that while linear variations yield straightforward averages, finding the average of polynomial functions requires integral calculus, where the average is determined by integrating the function over a specified range. A specific issue arises with the charge density of a sphere that varies as βt, where the average does not yield expected results, prompting questions about the method used. The conversation emphasizes the importance of understanding the underlying principles of averaging, particularly in non-linear contexts. Overall, the thread seeks clarity on applying these concepts to more complex functions.
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Now we've all been taught how to use the average. Let me give 2 examples to those who don't know.

Example 1: Say an object moves with velocity 3t in the time t=0 till t=2. Find distance covered.
Initial velocity = 0.
Final velocity = 6 disp. unit/ time unit.
Avg. Velocity = 3 disp. unit/ time unit.
Distance covered = Avg. velocity x time = 6 disp. units.
Using s = ut +1/2 at^{2} we get 6 again. Amazing!

Example 2: Force acting on a box of mass 1 unit is 3t in the time t=0 till t=2. Find work done by the Force. Box is initially at rest to your frame.
No other forces act on it.

Initial force = 0
Final force = 6 units.
Avg. force = 3 units.
Now avg. accn. = 3 units [mass = 1]
As in previous sum, displacement = 6 units.
Work done = 3 x 6 = 18 units. This comes out fine if you work it out the normal way also.

Now onto my questions.

If you noticed both were linear variations. How do I find the average of any polynomial function? I would find that VERY useful. For instance I found out for a cos/sin function average is 1/\sqrt{2} of the co-efficient of the cos function. Isn't that fantastic?

Also one more. I was given a problem that the charge density of a sphere varies as \betat. But when I tried average, it doesn't work although it seems to be a linear variation.

Why doesn't it work?
 
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Can you expand a bit on the problem at hand?
 
You use integral calculus in the general case. Just like a simple average is the sum of the elements divided by the number of elements, a generalized average is the quotient of two integrals.
 
To calculate the average (also called mean), integrate the function over the range of interest and divide by that range.
 
berkeman said:
You use integral calculus in the general case. Just like a simple average is the sum of the elements divided by the number of elements, a generalized average is the quotient of two integrals.

For example, see the end of this:

http://math.cofc.edu/lauzong/Math105/Section%205.4%20Applying%20Definite%20Integral.pdf


.
 
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I see. Thats awesome! Average is such a nice way of going about the problem. What about the sphere of charge? Why can't I average that?
 

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