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

Averages of curve you learn about in Calculus through 2 new math operations. You learned multiply, divide, add and subtract, but there are 2 more. The operators which are used when numbers are constantly changing and not in a linear away. Enter Sir Isaac Newton. He was observing the motion of the planets and noticed you could not use any existing math to explain it, so he invented Calculus and the 2 operation, derivative and integral which operate on functions, not numbers.

Consider the path of a projectile. It follows a parabola, a curve. If I were to ask you at 2.5 seconds, how fast is the projectile coming down, with normal math you would be hard pressed to figure it. You could but it would a complicated process. With Calculus and the derivative, you take your function, the path of the projectile and take the derivative of it and now you have a function for the rate of change for the path. You could take the derivative again and get the rate of change, of the rate of change. We are essentially going from Position of the projective, and what do you call a change in position over time? Velocity, and what do we call a change in velocity over time, acceleration. So acceleration is the 2nd derivative of the function for the position of the projectile and integral is the ‘opposite’ of the derivative. If you have a curve, The integral of that curve is the sum of the area underneath the curve. It just so happens when you add up all the acceleration changes, you get the formula for the velocity, and when you add up all the velocity changes, you get the formula of the position.

So, the average of a curve would be to take the integral of the formula of that curve and then divide it by the difference in the starting and ending points of the curve, which looks like the top formula.

[https://calcworkshop.com/wp-content/uploads/average-value-theorem.png](https://calcworkshop.com/wp-content/uploads/average-value-theorem.png)