# Using average

## Main Question or Discussion Point

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 a$$t^{2}$$ 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 $$\beta$$t. 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?

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

berkeman
Mentor
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 [Broken]

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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?