I didn’t pass the Integrating Multivariable Functions unit test this week. 😔 As I mentioned at the end of my last post, I was fairly confident going into this week that I’d get through the unit test, so I’m pretty disappointed that I didn’t. The silver lining is that on average I was getting something like seven or eight questions correct in a row before getting one wrong, so I at least have that going for me which is nice. However, a lot of the questions I was getting correct was because I had memorized how to solve the questions but still didn’t really understand why the math worked. That said, I can tell I’m slowly starting to get a clearer picture of what’s going on. For example, I have a much better understanding of the gradient of surfaces and it’s relationship with the slope of each variable (for example, I can visualize the derivative of x with respect to z, or the derivative of y with respect to z). I’m starting to intuitively understand the difference between scaler fields and vector fields which, after literally MONTHS of studying the two, is certainly a relief. So even though I didn’t get through the test, which I’m legit super disappointed about, I can tell I’m making progress which I’m happy about.

Here are seven questions I worked through this week:

Question 1

(FYI, I worked through this question on my iPad which is why the screen shots look different than usual. I also may have missed screen-shotting the top part of the question, but I can’t remember or tell. 🤔)

This is a good example of a question that I have more-or-less memorized the how of how to solve it and have a 4/10-ish understanding of why the math works. I can now pretty much visualize how AND why a sphere is denoted with the notation shown in the first few steps of KA’s answer. I understand that theta is the angle used for the azimuth plane (which is the fancy word for the (x, y) plane) and that phi is used for the angle running up and down the z-axis. I can also visualize how AND why the x- and y-coordinates are dependent on both angles and the z-coordinate is only dependent on phi. I don’t understand, however, why you need to use the Jacobian to scale the Cartesian coordinates to spherical coordinates, nor do I understand why the linear algebra works. So, even though I got his question correct and am fairly confident I will get through these questions relatively easily next week, I don’t really know what’s going on which is, again, disappointing to say the least. 😒

Question 2

I got this question wrong because I forgot that I needed to add in the Jacobian into the integrand. Again, I don’t understand WHY the Jacobian needs to be added… But I now find it pretty straightforward to calculate the integral which is nice. The how of how to solve these questions is getting pretty easy for me which is definitely a positive thing.

Question 3

As you can see from the bottom of my note, I didn’t know what I was doing but I managed to get the question correct. This is a pretty good example to show that I’m getting much better at working through the determinants, the substitutions, and the integrals of these types of questions. 😁

Question 4

I worked through a number of these types of triple integration questions this week (and last week for that matter) and, for the most part, didn’t feel the need to add them to this post. This question in particular, however, took me THREE days to get through… (Although, to be fair, I’m pretty sure I only spent ~30 mins on it the second day.) The calculus wasn’t too difficult, but I kept making careless mistakes over and over when using the reverse-power-rule and trying to simplify the fractions. This type of question is a good example of how the math in certain questions isn’t too difficult but when there are 10 – 15 steps in a question, it’s very easy to screw something up a long the way and so the question becomes SUPER difficult…

Question 5

I’m doubtful anyone could interpret my notes from this question, but I wanted to add it here simply because I was pumped that I was actually able to make sense of the question and knew what to do to solve it. I definitely need more practice, but I can now pretty quickly decipher this type of double integral and visualize what I’m looking at. I could tell right away that based on the ((9 – (y – 1)²)^1/2 in both the bounds of the inner integral, the shape in question was a circle. I also knew that the center would be shifted up to y = 1 from those expressions on the bounds of the inner integral, and that it would also be centered at x = 1 based on the bounds of the outer integral since they ranged from –2 to 4. So, ya. I’m pretty good. 👊🏼 😤

Question 6

This was one of the types of questions I got wrong last week because I forgot how to solve them. I remembered the process pretty quickly this week but it did take me a moment to think through. This type of question is one of the rare ones where I actually understand the why of why you need to add the H(y) and G(x) functions to the end of each antiderivative. I explained it in my last post so I’m not going to bother reiterating it here, but I’m pumped that I understand it. ☺️

Question 7

As you can see at the bottom of my note, I got this question wrong but would have gotten it correct if I’d added the brackets to the first expression in the integrand. That said, I struggled a LOT with the exponent rules in this question. It’s embarrassing to say but I was having a hard time remembering the negative and fractional exponent rules and had to spend ~10 minutes working through examples I came up with on my own to remember how they worked. On the plus side, I knew how to solve this question (or, at least, I knew how to do the calculus in it…), I just forgot to add the brackets so that the Jacobian would be multiplied across both terms in the binomial.

I am PRAYING to the math gods that this week I FIN-A-LLY get through this unit test. 🙏🏼 🙏🏼 🙏🏼 I think I’m getting close to 20 weeks working through this unit which is up there with the longest amount of time I’ve spent working on any unit. (I feel like it might already be in the top 10.) Not to mention that this unit has been a battle trying to understand. I somehow doubt the following unit, Green’s, Stokes’, and the Divergence Theorems (0/600 M.P.), is going to be much easier, but when I finally manage to get there, it will just be nice to feel like I’m making some progress for the first time in ~5 months. 🥵