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

The way to think of the average height of a line, or a curve, on a graph is to think about the area under the curve. The average height is the height of a horizontal line starting and ending at the same x-coordinates as the line or curve in question which has the same area beneath it.

For a straight line segment, that will always be the y-coordinate of the point in the middle of the line segment.

This would be much easier to explain with a drawing, but I don’t know how to add one here.

Draw a (sloped) line segment on a graph. Draw vertical lines at the x-coordinates that are the left and right ends of the line segment. Then draw a horizontal line through the midpoint of the sloped line, beginning and ending at the same x-coordinates. Notice that there is a triangle included under the sloped line which is not under the horizonal line, and another triangle under the horizontal line that is not under the sloped line. Notice that those two triangles are identical in size. You can see in that way that the area under the horizontal line is the same as the area under the sloped line.

The branch of mathematics that addresses problems like these is calculus. We’d have to get into that to explain why the area under the line (or curve) is the way to think about the average, as well as how you can compute the area under something that isn’t a straight line. I don’t think I can ELI5 calculus.