I’m a bit disappointed with how my week went this week. I think I just made it to 5 hours of studying calculus in total but it was close. I felt a bit out of my depth this week with what I was working through which I’m guessing is part of the reason why I didn’t spend more time working on KA than I did. I only got through 1 section this week called Definite Integrals of Common Functions which contained 7 videos and 3 exercises. Although I didn’t get as far as I wanted, I did get a lot of useful practice in taking definite integrals back to their derivatives. (I’m pretty sure I’m saying that right but not 100% certain.) Spending the entire week working on these questions helped make integrals seem a bit more clear to me so, overall, I don’t feel too bad with what I was able to learn.
Every question I worked on this week followed the same general process; I was given a definite integral (i.e. an integral with a lower and upper bound), then asked to convert it to it’s derivative (again, not sure if I’m saying that right), and then, using the fundamental theorem of calc, a∫b f(x)*dx = F(b) – F(a), find the area between the function f(x), its bounds, and the x-axis. Here’s an example:
- 3∫12 (6x + 1) * dx = ((6x1+1)/1+1) + (1x0+1)/0+1)
- = 6x2/2 + 1x1/1
- F(x) = 3x2 + x
- = (3x2 + x) 3|12
- (Note: the vertical line means to find the difference when inputting 12 and 3 into (3x2 + x) respectively and then finding their difference, i.e. using the fundamental theorem of calculus.
- = F(b) – F(a) = F(12) – F(3)
- = (3(12)2 + 12) – (3(3)2 + 3)
- = (3(144) + 12) – (3(9) + 3)
- = (432 + 12) – (27 + 3)
- = 444 – 30
- = 414
After going through a few videos and starting to get a better feel for the pattern/process of solving these questions, I was then shown how to solve them when dealing with a piecemeal function, a.k.a. a function that has a jump discontinuity. The process to solve (- or evaluate? I’m not sure -) these types of functions is the same, but you have to use the FTC on each section of function before and after the jump discontinuity and then find their sum. Here’s an example:
- f(x) = { (3(x)1/2) for (x > 4) ; 2x – 8 for x ≤ 4
- 3∫9 f(x)* dx = (3∫4 (2x – 8) * dx) + (4∫9 3(x)1/2 * dx)
- = (x2 – 8x 3|4) + (2x3/2 4|9)
- = (F1(b) – F1(a)) + (F2(b) – F2(a))
- = (((4)2 – 8(4)) – ((3)2 – 8(3))) + (2(9)3/2 – 2(4)3/2)
- = ((16 – 32) – (9 – 24)) + (2(27) – 2(8))
- = (-16) – (-15) + (54 – 16)
- = -1 + 38
- = 37
As you can see, there’s a lot of algebra involved with these types of questions and a number of separate calculations that need to be done individually. Not only did I have to figure out the pattern/process needed to solve these questions, but, because there were so many steps, I had to make a point of writing neatly so I didn’t mix up the terms and also take my time and not rushing through each question.
It took me until the end of the week to get through the 3 exercises in this section. The first 2 exercises I managed to get through on my first attempt but the third exercise took me 2 days to work through. Here are 2 more example questions and my notes for each one:
As I said at the top, I’m disappointed that I only managed to get through 1 section of the unit Integrals (2,080/3200) this week but, on the other hand, I do think I managed to make some good progress understanding how to flip from a definitive integral to its derivative. I still think I have a long way to go to feel like I have a strong grasp on integrals, in general, and a REALLY long way to go before I intuitively understand what’s going and visualize it in my head. Nonetheless, I’m happy with the progress I’ve made so far, even if it’s happening slower than I’d like it to.