I was close to reaching my goal this week of getting through the final exercise in the section Double Integrals and also of getting through the entire following section, Double Integrals (Articles). I made it through the exercise in the former section but only got through three of the four articles in the latter. Although I was close to reaching my goal, I’m pretty disappointed with my effort overall. I barely studied for five hours and could definitely have put in more time and effort. I’m still struggling to visualize why the math in double integrals works the way it does (which was a big part of my lack of motivation). Similarly to last week, through a bunch of trial and error I managed to ‘brute-force’ my way through the final exercise in Double Integrals but almost had a nervous breakdown as I worked through it from not understanding what I was doing. I have a decent grasp on what is going on with double integrals (ie I understand the bigger, low-resolution picture of what’s going on) and am actually pretty good at doing the math, but I still can’t visualize why the math works, so I feel very lost. I know I’m going to figure it out soon (I have a feeling that double integrals will seem completely obvious once I do), but as of now I’m pretty demoralized. 😒
Before attempting the final exercise from Double Integrals, I started this week with the articles from the Double Integrals (Articles) section. I took a bunch of screen shots from the first article to summarize what it talked about, which you can see below. The first screen shot summarizes what a double integral does. The screen shots after that work through an example. You can see me notes working through this example at the bottom:
(INSERT SCREEN SHOT)
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(As I’m writing this point, I think I’m FINALLY starting to understand and be able to visualize what’s going on. In this example, as you’re integrating x from 0 ≤ x ≤ 2 you’re finding the width of the 3D object across the x-axis from 0 to 2. If you go “up” the y-axis and keep y constant at any point, say you hold y to equal 1, then the area of the “slice” of the object at y = 1 would be calculated with the formula 4 +2sin(1). The formula 4 +2sin(y) will tell you the area of every single “slice” from 0 ≤ x ≤ 2 if you hold y constant at any point. The next two screen shots finish off the double integral by integrating y from –π ≤ y ≤ π, meaning that you’re calculating the area of all the “slices” going up and down the y-axis from –π to π across 0 ≤ x ≤ 2 and then adding them all together… I think I explained that poorly, but I also think I finally understand how this works. 😤)
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I didn’t make any notes on either the second or third article that I thought were post-worthy. I did, however, grab a screen shot from the third article, Double Integrals Beyond Volume, which summarized the gist of that article:
Once again, I didn’t really understand what this article was talking about as I went through it. Up to that point, double integrals had been explained to me as finding the volume of a 3D object by first finding the area of “slices” across either the x- or the y-axis and then summing them together with the “slices” from the other, perpendicular axis. I believe the gist of what this third article was saying is that going forward, it will often be easier to visualize double integrals as calculating many, many columns in a specific region with a length and width of dy and dx. I don’t actually know if this is true though…
I first attempted the fourth exercise from Double Integrals early in the week but it took me until Saturday to get through it. I got rocked by it the whole time and it wasn’t until the very end of the week that I sort of got a grasp on what was going on. Here are two example questions from the exercise:
Question 1
When I first attempted this exercise, I didn’t realize I was going to have to draw out the different shapes which were given by the bounds of the integrals. Even once I realized that, I found it pretty hard to figure out what the shapes looked like. In the question above, trying to figure out that the top line of the triangle, which came from the upper bound 2 – 2(y – 1), was very difficult for me. Even once I figured it out, I still found the question super confusing in that it says, “Switch the bounds of the double integral” except the dxdy stays in the same order which seems to imply the bounds aren’t being switched… 🤔 Nonetheless, looking at the triangle I drew, I could tell that the y-axis would be going from 0 to 2 and the lower bound of the x-axis would be going from the bottom line of the triangle, x/2, up to the top line of the triangle, –x/2 + 2.
(To be honest, I still don’t understand how the answer implies that you don’t integrate beyond x < 0.)
Question 2
This was one of the last questions I worked on from the exercise and by that point I had a semi-decent grasp on how to solve the questions. (Ie I could do the math but didn’t know why the math worked.) It took me a really long time, however, just to figure out what the shape looked like in this question. Looking at the upper bound of the inner integral, 3 – 3x2, I knew that inputting x = 0 would output y = 3 meaning the y-intercept was 3. I also knew that the negative sign in front of the 3x2 implied that it would be an upside-down parabola. I couldn’t remember if the scaler 3 narrowed or widened the parabola though. I eventually figured it out and drew the correct shape. Since the initial double integral had y as the inside integral, I knew the solution would have the range across y on the outside integral of the solution and that it would go from 0 to 3. In the top right corner of my notes you can see that I switched the upper bound from being in terms of x to being in terms of y which was the correct thing to do. I stupidly didn’t realize that when I took the root of 1 – y/3 the solution could have been (+) or (–) and so my final solution had the lower bound being positive when, based on the shape I drew, it OBVIOUSLY should have been negative… AGH!! 😡 I had to restart the exercise but then ended up getting through four questions correctly in a row after that and finally (FINALLY) finished the exercise.
I’m really hoping this coming week will be more productive than this past week. If double integrals weren’t hard enough, the next section is titled Triple Integrals… But it only has three videos, one article and one exercise in it. I actually already watched the first video too, so I don’t think it will take me too long to get through the section. I’m only 45% of the way through Integrating Multivariable Functions (720/1,600 M.P.) which is disappointing considering how long I’ve been working on this unit. 😒 I’m praying to the math gods that I’ll make some decent progress this week so I can at least another 5–10% of the way through it. 🙏🏼 🙏🏼 🙏🏼