After five long weeks, I FINALLY got through the Integration and Accumulation of Change unit test. 😭 I ended up passing it on my first attempt this week — I was already 15 questions into the test when I started it — and felt quite confident answering all but the final question. It was a HUGE relief finally getting through it. After that, I started the next unit test I need to get through, which is Unit 9 in Calculus BC, Parametric Equations, Polar Coordinates, and Vector-Valued Functions. Unfortunately, I got my butt kicked on pretty much all of the questions I attempted on it. The good news is that this unit test is only 14 questions long, so I don’t think it will take me too long to get through this one, at least not nearly as much time as the previous one did. But that said, I felt pretty lost on most of the questions that came up, so it may take me a while to review what I need to to regain my grasp on everything. Still, I’m PUMPED to be moving on from the Integration and Accumulation of Change unit test and feel like I’m back on track to finish KA before I turn 80 years old. 😮‍💨

I flew through the final 15 questions for the Integration and Accumulation of Change unit test so fast that didn’t bother taking any screenshots until the final question:

Calculus AB – Unit 6 – Integration and Accumulation of Change – Unit Test

Question 1

I cheated on this question by using Symbolab to double check my answer and it’s a “good” thing I did because I would have gotten it wrong. I knew that the integral of ∫1/(6² + (x + 4)²) would be arctan((x + 4)/6)² but forgot that I needed to put 1/6 out in front. If I hadn’t checked Symbolab, I would have gotten it wrong which is disappointing, but in the truth is that I don’t intuitively understand the proofs for any of the arc-trig formulas I’ve been using, so in that sense I’ve been “cheating” the whole time using formulas that I don’t completely understand. 🙁

After I finished that unit test on Tuesday, I moved on to the Parametric Equations, Polar Coordinates, and Vector-Valued Functions unit test which is Unit 9 of Calc BC. Here are four questions from that test:

Calculus BC – Unit 9 – Parametric Equations, Polar Coordinates, and Vector-Valued Functions – Unit Test

Question 2

I got this question wrong but would have gotten it correct except I didn’t see the sin in front of y = sin(4t – π /2). That said, I had no clue what I was doing and just assumed I needed to find the derivatives of y’(t) and x’(t) and divide them by each other. That turned out to be correct, so I would have gotten it correct if I didn’t misread the question. 😒

Question 3

I spent about 30 minutes looking at this question trying to remember what to do but couldn’t figure it out. I eventually gave up and looked at what KA did to solve it. I was still pretty confused, so I watched the video in their answer:

The video makes it pretty easy to understand the formula for the integrand used to find area between the origin and the curve, but I was still having a hard time understanding why the shape of the curve in the question (the swirly thing) looked the way it did given the function r = θ – 1. I ended up finding this video which helped a bit:

This vid helped me remember that every point along the swirl represents the length of the radii as you move along the unit circle. So, at π, the distance from the origin to the first ring of the swirl at angle π is ~3.14. At 3 π, the distance from the origin to the second ring of the swirl at angle π is ~9.44. (That probably doesn’t make any sense but I don’t know how to phrase it…)

In any case, I didn’t really figure out why the swirl in the question looks the way it does given the equation r = θ – 1 or understand why the lower bound of the integral equals 1, but I at least understand the integrand, so I’ve got that going for me which is nice.

Question 4

I also spent about 30 minutes trying to figure this question out but was once again stumped.  I didn’t remember that x(θ) = r * cos(θ) or that given r(θ) = [some function], that the derivative at a given x-coordinate of x(θ) would be d/dθ [r(θ)x(θ)] = r’(θ)x(θ) + r(θ)x’(θ). The good news is that I don’t find using the chain rule or doing this type of math very difficult so, again, I’ve at least got that going for me which is nice.

Question 5

I actually got this question correct but only because I knew that d²f/dt²[t⁶ – 9t³] = 30t⁴ – 54t which was only a part of option C, so I got it correct but was annoyed that I couldn’t remember why the second derivative of log₄(t) = 1/t²ln(4). I watched this video to try to remember:

As you can see, the algebra, log rules, and calculus used for this proof is pretty easy, but I couldn’t remember why d/dx[ln(x)] = 1/x so I tried to figure it out watching this video:

I unfortunately didn’t understand this video though. 😞 I feel like the derivative of ln(x) is a pretty key derivative to understand so I may try and watch a few videos on it this coming week to hopefully wrap my head around it. In any case, this was the last question I worked on this week and was the first one I got correct on this new unit test. 💪🏼

Like I said, I’m PUMPED to be done with the Integration and Accumulation of Change unit test and am optimistic that I’ll be able to get through Parametric Equations, Polar Coordinates, and Vector-Valued Functions unit test relatively quickly. I think it will take me a bit of time — maybe two or three weeks? — to review everything in order to be able to get 14 questions correct in a row, but I don’t think it will be too bad. I looked at how many more unit tests I have left in the High School and College Math section of KA and given how many there are, it will be pretty much impossible for me to finish them all before the start of the New Year. So that’s disappointing, but the finish line is much more clearly in sight now and I know I’m going to get there. 🏎💨🏁

As always, fingers crossed that I can make some decent progress this coming week so I can get there sooner rather than later. 🤞🏼