Averaging Polynomial & Non-Linear Functions: Examples & Explanations

[WiFO215]

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?

 

[swapnilster]

Can you expand a bit on the problem at hand?

 

[berkeman]

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.

 

[marcusl]

To calculate the average (also called mean), integrate the function over the range of interest and divide by that range.

 

[berkeman]

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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[WiFO215]

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?