I put in a solid effort this week on KA and made some decent progress. In terms of achieving Mastery Points, I’m now 10% further through Integrating Multivariable Functions than I was at the start of the week. I’m pretty happy with that, but I only made it through three videos, two articles and two exercises so although the M.P. I achieved was decent, I didn’t actually make it through that much content. The majority of my week was spent working on a single exercise from the Triple Integrals section. The math from this exercise wasn’t too difficult, but each question had 10 to 15 steps to work through and I kept making careless mistakes along the way. It was frustrating for sure but also good practice working on integrals and actually kind of fun. 🤓 All in all, it ended up being a pretty good week even though I think could have (and wish I would have) gotten further. 🤷🏻‍♂️

I started the week reading through the final article in the Double Integrals (Articles) section which was titled Double Integrals in Polar Coordinates. I always found polar coordinates a bit tricky, in general, so I was worried heading into this article. It turned out that not only was I able to understand the polar coordinate system fairly easily, I was able to understand how to calculate double integrals using polar coordinates AND had a decent grasp on why the math works. Here are a few screen shots from that article which summarize most of it:

To reiterate what these screen shots explain, the first step to take to understand how/why using double integrals to calculate the area of a shape with polar coordinates is to visualize chopping up the shape (in this case a circle) into a bunch of tiny little rectangles. The length of any given rectangle on the circle is dr (a.k.a. a tiny, tiny nudge in the direction of the radius) and the width of any given rectangle is dθ (a tiny, tiny change in the angle). I think I understand this part clearly, but I’m not 100% clear on why you need to multiply the rectangle by r (the radius up to that rectangle). I think it’s because if you’re calculating the width of a rectangle closer to the center of the circle compared to one near the perimeter, the same change-in-angle, dθ, will result in two different widths for each respective rectangle so multiplying both rectangles by their respective distances away from the center (ie r) will account for this. As usual, I’m not actually sure if that’s true though.

On Wednesday I started the Triple Integrals section. It contained three videos, one article and one exercise. I had already watched the first video and decided to jump ahead to the article in hopes that reading through it would make the videos a bit easier to understand. The gist of triple integrals is that they’re the exact same as double integrals but (as you’d expect) add in a third dimension, z, so that instead of calculating the area of a 2D shape, you calculate the volume of a 3D shape. (I’m guessing there are more applications of triple integrals than that, but that’s all I learned about this week.) There’s not much more to add to explain triple integrals, so below are four example questions I worked through. The first question was from the article and the following three were from the exercise:

Question 1 

Question 2

Question 3

Question 4

As I mentioned in my intro, the exercise was pretty difficult for me to get through not because the math was tricky – as you can see from the example questions, most of the functions in the integrands had coefficients of 2 or 3 and the bounds were often 0 or 1 – but I would often get the questions wrong since there were so many steps and I’d make a careless mistake somewhere along the way. It was frustrating at times, but I enjoyed working through this exercise as it felt like solving little puzzles and was super satisfying when I ended up getting it correct and the solution would be some random fraction.

The next section (which was where I ended this week) was titled Change of Variables and oddly only contained two exercises with no videos or exercises. I flew through the first exercise and opened the following exercise to look at the first question but didn’t have time to start it. The first exercise was the easiest thing I’ve done on KA in a LONG time. It was to the point where I’m not sure if I really understand what the purpose of the exercise was… 🤔 All I had to do was take a two input, one output function, ex. f(x, y), and replace the variables x and y in the function with other expressions and then simplify the function. It was pretty basic algebra that I learned how to do years ago which is why I feel like maybe I missed the point of the exercise. Also, I just asked CGPT what the name of this type of two input, one output function is and it said it’s called either a “binary function” or two-variable function. I will henceforth be calling them binary functions. 🤓 Anyways, here are two example questions from that exercise:

Question 5

Question 6

I’m now 55% of the way through Integrating Multivariable Functions (880/1,600 M.P.). I feel like I’m getting pretty close to finishing off this unit, but there’s a section coming up that has THIRTEEN videos in it… 😳 Apart from that, the remaining sections only have two to four videos, articles and/or exercises in them so they should go by pretty quickly. (Hopefully.) This unit feels like it’s taken me forever to get through but I’m pretty optimistic about finishing the rest of it off. I think it’s possible that I could get through it before the end of the year, but it could be pretty tight. As always, fingers crossed I can make some solid progress this coming week. 🤞🏼