Why does dividing the final(chosen) height in a line by 2 give you the average height(y) in that line?

Question

I was taught, of course, that the average of something is the sum of all the parts divided by the number of those parts.

This is easy when it is a set of values(2, 6, 8) because its simply (2+6+8)/3 = 16/3 ≈ 5.33.

But when you have a line, you have an **infinite amount of values** divided by **infinity**, right?

So how is it even possible to take the average? And why can you simply divide it by 2 when its a line?

And then of course there is the question about the average of curves and whatnot…

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I feel so dumb, plz help me understand.

**Also this question comes from the idea that average velocity is half the final velocity**

In: 0

Answer 1

You can think of average as “with what number do I need to replace all the numbers for the result to stay the same”. 3 + 6 + 12 = 7 + 7 + 7. This obviously depends on the operation, for adding it’s the arithmetic mean, for multiplying it’s the geometric mean (3 × 6 × 12 = 6 × 6 × 6) etc.

This generalizes fairly reasonably to the continuous case. If a car’s speed varies over time but by the time *t* it covers the distance *d,* then the average speed is the constant speed that, if maintained at all times during the period of time *t,* would cause the car to get to *d,* and that’s easy to calculate, *d/t.*