My final week of my third year working on KA was unfortunately not the most productive. I had a few things come up this week that got in my way of working through KA. On Monday I had a big–ish meeting with my boss which I spent the entire morning preparing for, on Saturday I spent the morning getting ready to leave to coach at a squash tournament all day, and from Tuesday to Thursday I didn’t have the internet because my modem died. (I used data from my phone but it was slower going than working on my computer.) Nonetheless, I still got some decent work done although I still didn’t passed the Applications of Integrals unit test. 😡 I spent the entire week going through the test twice and working through ~30 questions. I didn’t really learn anything new so this entire post is just going to be 5 example questions that I worked through. Fingers crossed the first week of my fourth year (😲!) working on KA will be more productive than this week was.

Here are the five questions:

Question 1

The note you see above is one that I just rewrote and I’m not exactly sure what I did the first time I worked through this question or why I got it wrong. Looking at it, I’ve found these types of questions a bit confusing in the past even though it doesn’t seem all that difficult to me now. This is a pretty straightforward question dealing with the F.T.o.C. so what this tells me is that I still need more practice using this theorem to get it better ingrained in my mind.

Question 2

I was annoyed that I got this question wrong. I actually completely understood that r(t) represented the derivative as the question stated it was the “rate of change” or liters/hour. I knew that meant that if r(t) was in the integrand it would output the total amount of liters produced from ₂₀∫²¹ but I answered that it was in the 20^th hour forgetting that the first hour, n = 1, would have been from t = 0 to t = 1. 

Question 3

It took me about 5 minutes to think through this question and I was happy that I was eventually able to figure it out. The part that gave me a bit of trouble was understanding how to use arctan(x). After ~2 years of working on trig questions, I still don’t have a completely firm grasp on arc-trig functions which is disappointing. Going forward I need to always remember that the inverse of sin(x) = opp/hyp is simply x = sin^-1(opp/hyp) and the same applies for cos(x) and tan(x). In any case, I eventually worked it out and was able to remember the values for 30-60-90 triangles to get the correct solution.

Question 4

This was another question where I didn’t understand that I needed to use the basic F.T.o.C. formula to get to the solution. I tried solving ∫ ln(t + 1) sin(t) dt using integration by parts which was impossible for me. I also had a feeling I was doing it wrong since the question asked me to use a graphing calculator. When I looked at the answer given by KA I realized how simple it was and, seeing how to properly solve this question, I think it helped me to get a more solid understanding that _a∫^b f(x)dx = F(b) – F(a).

Question 5

The main reason why I got this question wrong was because I forgot to subtract 1 from r = y^1/2 + 1 which you need to do because the area being calculated, “R”, is being rotated around the axis x = 1. As I rewrote this question in my notes, I realized that I still find these types of questions hard and need more practice with them. In general, I think I understand WHY this concept works the way it does but, because there are so many steps in each question, I need to get better at working through each step slowly to get to the correct solution.

This coming week marks the beginning of my FOURTH YEAR (!!!) working on KA. Technically the exact 3-year mark will be this coming Friday, Sept. 2^nd so I’m hoping by then I’ll have passed the unit test for Applications of Integrals (1,740/2,000 M.P.) so that I can move on to the following unit Parametric Equations, Polar Coordinates, and Vector-Valued Functions (0/1,500 M.P.). I need to redo 2 exercises from Applications of Integrals before I can restart the unit test so getting started on the following unit may or may-not happen by Sept. 2^nd. Either way, I’m getting closer and closer to finally (FINALLY) getting through calculus. And it’s only taken me three years… 😳