New video: Averages with integral, EoC Chapter 9

Another way to think about the average of height of a function is to think of a rectangle with the same area as the integral and the same base as the interval. Imagine pouring the area under the curve into this rectangle like water, and the water settles at the average height. So BH=integral. So H=integral/B. This also works for the average value of a surface over an area (think of the surface of the water in a chaotic bathtub over the base of the tub. Eventually the water settles on the average value. In this case the integral would be the volume under the surface rather than the area under the curve).

Chuck Larrieu

Great problem, classic problem, though I'm not sure I'll be able to squeeze it in here. Perhaps at a later date.

3blue1brown

I remember a similar feeling the very first time I saw this view, which is probably why I wanted to put it in the series :)

3blue1brown

Another great video! Yes, you did repeat the explanation. But the first time you showed an example and then you generalized. I think it's perfect, it helps solidify the ideas. It wasn't too repetitive for me.

When you first framed the idea of an average of a function in terms of the average slope between the two points, my head exploded it was so cool. Thank you for making it easy to understand cool math. And with videos like this series, thank you for creating a brand new understanding for things I've already learned, it's bloody fantastic.

Mr. IntelliGent

Maybe using frequency would help in converting discrete average to continuous average. In the last video, you introduced integral as the sum of areas, each area being a prespecified width times the height of the function at a point. So when summing all the points, you could assign all the points on a subinterval to have the same function value. To be technically correct, you could use underestimators and overestimators; and show through sandwich theorem that the integral is the correct value to attribute; though that may get too complicated.

Magnasium

Might you want to put the problem into this series: "if we throw a needle of length 1 onto lined paper with spacing 1 what's the probability it hits a line"? And no I didn't find it repetitive.

Jacob Mirra

Love the videos. Small comment though - when the emails get sent out clicking on the watch starts the video a few seconds in rather than at the start. So I miss the awesome quote unless I manually rewind. Feedback on the video itself - I learned all this stuff almost 40 years ago, and for the most part I understood it then. Your videos are like seeing the earth from space, sure - I'd travelled Australia, but it certainly is awesome to see it in context.

Christopher Burke