I reached my goal this week of getting through the Surface Integrals (Articles) section. I also worked through the final four videos in the previous section, Surface Integrals, which were a mini-series that worked through a single question (which FYI is the first question below). The latter section helped me understand the difference between a ‘surface AREA integral’ and a ‘surface integral’ (although I had to use CGPT to really wrap my head around it). Of those four articles, two of them were dedicated to working through a single question and which were both were long and arduous. I’m really pleased with the effort I put in this week and feel like I made some solid progress. It’s officially January 1^st so I didn’t reach my goal of getting through this unit before the end of the year, but I’m still on the right track and am more motivated than ever! So I’ve got that going for me, which is nice.

Often I reiterate in these posts what I wrote in my notebook to make it easier to understand the questions I post about. A lot of times I feel like my notes alone wouldn’t make sense without more context and/or a bit of an explanation. However, I think this week I actually did a pretty good job of working through the questions step by step and showing how each step worked in my actual notes (and also talking about the parts that I didn’t understand). So I’m not going to bother reiterating the process of solving each question this week as hopefully my notes will be clear enough on their own. 🤞🏼

Here’s a few screen shots from the final four videos in the section Surface Integrals and my notes solving that question below. As you’ll see, the question worked through solving the surface area of an object that looked like a cylinder that was sliced off at an angle:

Question 1

The first two of the four articles in the following section, Surface Integrals (Articles), worked through surface AREA integrals and the last two articles worked on SURFACE integrals (which are actually harder even though the name for them might seem otherwise). It wasn’t until the end of the week that I started to understand the difference between the two (I think). The former simply calculates the surface area of an object whereas the latter tells you, for example, the temperature of a surface (I think the average temperature, more specifically, but I’m not sure) where the temperature varies at different points along the surface. Here’ CGPT’s explanation of the difference between the two:

With that explanation out of the way, here’s a bunch of screen shots from the first article and my condensed note working through the question from it:

Question 2

The second article was more of the same. Here’s a bunch of screen shots from it and my notes:

Question 3

As I mentioned, the third and fourth articles got into how to calculate things like temperature variation across a surface. My ‘laymen’ understanding of the main difference is that a surface integral that calculates the temperature will need to incorporate the function FOR the temperature in the integrand of the double integral. (That may not be an accurate explanation.) I didn’t take any notes on the third article, but here are some screen shots from it:

Finally, here are some screen shots from the final article and my ntoes working through it below:

Question 4

Getting to the end of this question, I don’t really understand what the answer, 80π, means. I think it means something like, if a function for the temperature on the surface of the given sphere was equal to –2x + 5, the total temperature across the surface would be equal to 80π. That’s confusing to me though and I feel like that’s not actually what the solution 80π means… 🤔

I only have two more sections left in Integrating Multivariable Functions (1,280/1,600 M.P.) which are titled Flux in 3D and Flux in 3D (Articles). Like I said at the end of my last post, I’m hoping and think I can get through both sections this coming week as the former only has three videos in it and the latter only three articles. Assuming I do, I also think I should be able to make it through the unit test at least once this coming week. I have a feeling I’m not going to pass it on my fist attempt (I’d be VERY surprised if I did), but nonetheless, that will set me up well to get through it by the end of next week. Whenever I can manage to get through the test, I’ll FINALLY be onto the fifth and final unit in Multivariable Calculus, which is going on FIVE YEARS in the making. I’m pretty proud of all the work I’ve put in, but holy crap this has taken me a long time… 😅😮‍💨