This week was one of the most productive weeks I’ve had in a very long time. 😊 I began the unit Integrals and made it 13% of the way through by the time Sunday rolled around. (Actually, I had watched the first 3 videos in the unit last week but hadn’t made any notes until this week.) Everything I’ve learned so far has been straightforward and easy to wrap my head around. The majority of my week was spent learning about Riemann Sums but I also learned what definite integrals are and finished the week working through summation/sigma notation. I also (FINALLY) passed the Algebra 1 course challenge for Schoolhouse.world which means I will soon be certified to tutor Pre-Algebra AND Algebra 1. 🥳
The first thing that was covered at the start of Integrals was the notation for what Sal specifically referred to as “definite” integrals. In my mind, this implies that there are other types of integrals that I haven’t learned about yet. There were only a handful of videos on the definition for definite integral notation, however, so I still don’t have a much of a grasp on how/why the notation works. Here’s a page a from my notes that sums up what I’ve learned up to this point:
You can see the notation for a definite integral to the right of the graph which is a∫b f(x)d(x). As I wrote below the notation, the notation refers to the area between the function f(x), above the x-axis, and between the left- and right-bound, a and b respectively. I was shown the notation but wasn’t taught how to use it or how it works which I’m guessing I won’t learn for awhile. I’m also assuming that the purpose of being shown it now is just so that I get comfortable with how it looks and the names for the different parts of the notation.
Next, I learned about Riemann Sum Approximation. The general idea of RSA is that you can find the approximate area underneath a function by creating a bar graph which aligns with the approximate width and height of the function at specific intervals and then calculating the area of each bar:
In the above example, you can see that that width of each bar being measured is 0.5 which is labelled as Δx = 0. This means that the change in x for each interval is 0.5. Since the area for a rectangle is length * width, the area for each bar is xn * f(xn). Since you need to add all the bars’ area together, you can factor out the delta x which leaves you with the formula:
- Area = Δx(f(x1) + f(x2) + … + f(xn)).
When calculating the length of each bar (i.e. f(x)), the total area of the RSA will be different depending on if you set the top left corner of each bar to touch the function, the top right corner of each bar to touch the function, or the midpoint of each bar to touch the function:
(Note: In the photo above, there’s no example for a midpoint RSA.)
If the function is increasing (as it is in the above photo), taking a left-RSA will be an underestimation whereas taking a right-RSA will be an overestimation. Conversely, if the function is decreasing, a left-RSA will be an overestimation whereas a right-RSA will be an underestimation.
| Direction | Left Riemann Sum | Right Riemann Sum |
| Increasing | Underestimation | Overestimation |
| Decreasing | Overestimation | Underestimation |
It’s also possible to calculate an RSA where the widths of each interval (a.k.a. the subdivision/partitions) aren’t uniform:
When calculating an RSA where delta x is different for each bar, you must calculate the area of each bar individually and sum their individual areas as opposed to factoring out the delta x and then multiplying it by the sum of f(x).
The last thing I learned about RSA was how to use trapezoidal bars to calculate the approximate underneath the function. This is essentially the same thing but, instead of using bars (i.e. rectangles), you use right trapezoids to get a better approximation of the function’s area. The key thing to remember in order to do this is the formula for the area of a trapezoid:
- Area of Trapezoid = ((b1 + b2)/2) * h
- “Base-1 plus base-2, divided by 2, times the height.”
In the case of trapezoidal RSA, b1 and b2 are f(xn) and f(xn+1) respectively and h = Δx. Here’s a photo from my notes that explains this idea and a few screen shots of a question I worked through which also gives this concept more detail. (Also, please note that the formula with the “(?)” pointing at it in my notes is wrong. I was trying to come up with the formula on my own but I didn’t get it right.):
As an example, to calculate the area of a trapezoidal RSA with 5 trapezoids where Δx is equal for each trapezoid (as shown in the photo of my notes above), the formula is:
- Area = (Δx/2) * (f(1) + 2(f(2) + f(3) + f(4) + f(5)) + f(6))
Lastly, here’s an example from my notes of a trapezoidal RSA question I worked through:
On Saturday, I finished the week by learning about Summation/Sigma Notation. I’ve covered sigma notation before when I worked through Statistics but I still found it useful an informative to go over it again this week. Generally speaking, summation/sigma notation is used when you need to find the sum of a sequence of expressions, where each expression is identical to the one before it except for that one of the variables is being increased by 1 in each successive expression. (I’m fairly sure that doesn’t make sense but I can’t find the words to explain it properly.) Here are the notes I took about sigma notation which do a better job explaining how it works:
Before I wrap up this post, I want to mention how pumped I am that I finally passed the Algebra 1 Course Challenge for Schoolhouse.world! Pending it’s review and me assessing 2 other people’s applications, this means I will soon be certified as a Pre-Algebra and Algebra 1 tutor. I didn’t mention it in my last few blogs, but I’ve been trying multiple times per week to pass that course challenge and kept making stupid mistakes and failing. I was getting incredibly frustrated and demoralized which is why I’m SO happy to have finally gotten through it. WOO! 🎊
I’m now 13% of the way through Integrals (400/3200 M.P.). I’m happy with my progress so far but I expect that the next few weeks won’t be as straightforward as this past week. If I somehow manage to keep going at this same pace, I would be able to get through this unit before Christmas so, even though it likely won’t happen, that’s my goal for the time being. We’ll see! 🤞🏼🎅🏻