I got my ass handed to me this week on KA. I made it through a grand total of 1 exercise and 2 videos… That said, I just counted my notes and wrote out 42 pages worth of notes/equations so, even though I didn’t make much progress through Applications of Integrals, it wasn’t for a lack of effort. I got a much better understanding of how to use calculus to determine the volume of 3D objects, although I’m definitely I’m still far from fully understanding how it works. Nonetheless, even though it’s taking way longer than I want it to, I can tell I’m making progress and will eventually figure it all out. Eventually… 🙄

I started the week working on an exercise from the section Volume: Squares and Rectangles Cross Sections. I made it through that exercise on Tuesday and began the following unit, Volume: Triangles and Semicircles Cross Sections, on Wednesday. As you would assume based on their titles, these two sections worked on the exact same concept with the only difference being the shapes that were used in the calculations. Here are 5 questions between the two exercises from both sections that I worked on. The first two questions are from the ‘squares and rectangles’ section and that last 3 questions are from the ‘triangles and semicircles’ section: 

Question 1

(Usually I would take the time to rewrite the example questions here because it helps me to better understand them. I’m tight for time this week, however, and, since I also made a point of writing my notes out as legibly as possible, I’m not going to bother typing out any of the questions so suck it.)

Question 2

Question 3

(Note: In this question, the key thing to realize is that the ‘height’ of each triangle (i.e. it’s z-value) is e^-x/2. This is because it’s given that the shapes z-coordinate is an isosceles right triangle where the base is e^-x and, as shown in the top left corner of the first page of my notes, its height is therefore e^-x/2.)

Question 4

Question 5

1. Draw out the shape and label a point along f(y) as (x, y). You need to do this for step 4 in order to find the general equation of f(y).
2. Determine the ‘height’ (i.e. the z-value) of the shape in terms of x. Since the shapes ‘height’ is an equilateral triangle, the height/z-value is √3x/2.
3. Determine the equation for the area of each triangle ‘slice’.
4. Do some algebra to come up with an equation for x in terms of y which you need to do because you’re measuring the shape’s volume from y = a to y = 0.
5. Determine the equation for the volume of the shape.
6. Use calculus and algebra to find the exact equation for the volume of the shape given the variables a and b. 

After writing out >40 pages of notes this week, I’ve gotten a much better understanding of HOW to use calculus to find the volume of shapes (although I still need a lot of practice) but I don’t have much of an understanding of WHY it works… It feels like I have these seemingly disparate concepts floating around in my head that actually DO relate to each other but I just can’t visualize how they all simultaneously connect. I’m confident I’ll understand it eventually but I feel like it’s going to take a lot of practice to fully wrap my head around how it all works to the point where it seems clear.

I’m now 55% of the way through Applications of Integrals (1,040/1,900 M.P.). With 3 weeks left in May, I think it’s possible that I could get through this unit before the end of the month but it’s starting to feel less likely that I’ll also get through the Calculus 1 before the start of June. One good thing I realized is that, since many of the units/exercises are in both courses for some reason, I’m now 53% of the way through Calculus 2. My larger goal is to get through Calc 1 AND 2 before the start of September (i.e. the 3 year make of working on KA) so, if I don’t make it through Calc 1 before the end of May, I’ll still have a shot at achieving that second, bigger goal which is nice to have something to work towards.