Summer of Math Exposition

I think the video is very nice. The concept is easy to understand. The pre ration is very slick with the animated equations.

In the beginning when you are showing the area under the curve as the sum of the areas of the rectangles, you could make an animation where the number of rectangles slowly increases. In this way, you could show with the figure the effect of taking the limit of $dx \rightarrow 0$ resulting in a better approximation of the area under the curve. You spend a lot of time reading the coefficients of the functions you used to fit the profile of the vase, you can leave them on the screen without reading. On the other hand, it would be interesting to know the process you used to obtain those functions, right now they feel a bit magic and unexplained. Also you never define what “fit a function” means. When you introduce a disk method, you could represent the area of a thin disk as a rectangle of width $dx$ and height $\pi r^2$. You then show other disk to rectangle conversions and show that all the rectangles form an area under a curve. Right now the connection with the integration procedure you introduced in the beginning is not very clear. Around 7:10 you use for the first time the notation for the integration limits, but you do not explain it. You compute the relatively complex integral using the calculator, which again feels a bit like magic. I think using immediately the vase as an examples makes your explanation harder to understand. If you used some abstract vase with some simple functions as profiles you would not have to fit the functions and at the same time you could show how to perform manually some simple integration. After this abstract step, you can consider the vase and introduce the approximate fitting procedure. To perform the complex integral, instead of using the calculator, you could try to explain some simple integral approximation procedure such as Montecarlo. Calculating the stem volume as a disk integral is conceptually confusing. To use the disk method you assume to know the formula for the volume of a cylinder. Computing the volume of a cylinder with the disk method is a circular reasoning. You use the properties of the integrals without explaining why they work. In general, I don’t see the point of introducing the flowers they don’t add much intuition about what the integral is. Moreover, the actual measurement are a bit confusing since the observed volume WITH flowers is very close to the expected volume WITHOUT flowers. I certainly agree that this is due approximations errors, but this means that you do not have the sensitivity to account for the volume of the stem. In fact, the volume of the stem is 31 mL which is the same size of your approximation error of 30-40 mL.

This is a good explainer of the disk method. It gives a very clear example at the beginning and build to solving it at the end.

The investigation part showing the building and how you found the equations by trial and error. This could benefit after the point of the discovery of the formulas of more 3d graphics showing how the slices fit together and less equations on the screen.

The pace is fairly slow which will be good for high school or first year university students encountering the topic. However it probably would not be played “In the classroom” as it may not hold all students attention.