On one hand, I felt like this week was one of my most productive weeks so far (😊!!) but, on the other hand, I didn’t get very far through the unit Integrals. (😔) I spent the entire week working through one section, Riemann Sums in Summation Notation, and I was pretty much able to fully wrap my head around sigma notation (I think). By the end of the week I was surprised at how much of a huge relief this was. I find sigma notation pretty intimidating so getting a stronger grasp on it makes me feel a lot more confident moving forward into calculus. As I’ll talk about below, once I wrapped my head around the notation, I was a bit stunned at how much information is conveyed in sigma notation, especially considering how short it is. In retrospect, I suppose that’s the whole point of mathematical notation – to pack as much information as possible into the fewest characters possible. I just wish they didn’t look like hieroglyphs. 😵‍💫

To explain how sigma notation works, here are two example questions I worked through:

            Question 1:

(Note: The first page of my notes was my initial work answering this question and the second page was me writing the notation out more clearly and putting its meaning into words.) 

• Q. What is the sigma notation for a right-hand Riemann Sum of the function f(x) = (1/x) + 2 between the interval [2, 7] with 10 equal subdivisions?
  • Step 1 – Find Δx:
    • Δx = (b – a)/n
      • = (7 – 2)/10
      • = 5/10
      • = 0.5
  • Step 2 – Find the equation for x_i:
    • (Note: x_i means, “the x value of each term (?) in the index being calculated in f(x)“. As well, since Δx is equal to 0.5 between each of the 10 subdivisions, x_i is a linear equation. Because we know the ‘slope’ and the coordinates for 1 point (x₁₀ = 7, a.k.a. the 10^th number in the index has an x-coordinate of 7), we can use the point-slope equation to come up with expression for x_i.) 
    • y₂ – y₁ = m(x₂ – x₁)
      • y₂ = x_i
      • y₁ = 7
      • m = Δx = 0.5
      • x₂ = i
      • x₁ = 10
    •  x_i – x₍₁₀₎ = Δx(i – 10)
      • x_i – 7 = 0.5(i – 10)
      • x_i = 0.5i – 5 + 7
      • x_i = 0.5i + 2
  • Step 3 – Input the expression for x_i back into f(x).
    • f(x) = (1/x) + 2
      • f(x_i) = (1/(0.5i + 2)) + 2
  • Step 4 – Use sigma notation to represent the calculation of each rectangle in a right Riemann Sum:
    • Area = The sum of each of the 10 rectangle’s widths times their individual heights.
    • Area = _{(i = 1)}Σ¹⁰ W * H
      • = _{(i = 1)}Σ¹⁰ Δx * f(x_i)
      • = _{i = 1}Σ¹⁰ 0.5 * (1/(0.5i + 2)) + 2

(Note: Here are 2 more pages from my notes with the first page highlighting how the equation for x_i works and how it’s related to f(x) and the second page highlighting the difference between f(x) and f(x_i).)

            Question 2:

(Note: Because we’re calculating a midpoint Riemann Sum, it’s conventional for the index to start at 0.)

• Step 1 – Find Δx:
  • Δx = (b – a)/n
    • = (5 – 1)/4
    • = 4/4
    • = 1
• Step 2 – Come up with equation for x_i:
  • x_i – 4.5 = 1(i – 3) 
    • x_i = i – 3 + 4.5
      • = i + 1.5
• Step 3 – Input the expression for x_i back into f(x).
  • f(x) = x³/2
    • f(x_i) = (i + 1.5)³/2
• Step 4 – Use sigma notation to represent the calculation of each rectangle in the midpoint Riemann Sum:
  • Area = The sum of each of the 4 rectangles widths times their individual heights.
  • Area = _{(i = 0)}Σ³ W * H
    • = _{(i = 0)}Σ³ Δx * f(x_i)
    • = _{(i = 0)}Σ³ 1 * (i + 1.5)³/2
    • = _{(i = 0)}Σ³ (i + 1.5)³/2

As you can see from my notes on this question, although it’s not conventional, you could also state that the area = _{(i = 1)}Σ⁴ 1 * (i + 0.5)³/2 with the difference being that x₍₀₎ would start at 0.5 and therefore x₍₁₎ would then start at 1.5. I believe the reason why this isn’t the conventional method used is because doing it this way would set a outside of the rectangles in the Riemann Sum which would be a bit weird.

As a final side note, something that took me an embarrassingly long time to understand is that, even though the index in Riemann Sums represents the individual rectangles or trapezoids being accounted for, x_i represents the left, midpoint, or right x-coordinate of each rectangle/trapezoid. For some reason I found this concept really hard to wrap my head around. 

As I said at the beginning, even though I didn’t get very far through Integrals (480/3200 M.P.), I feel like the entire week was a good use of my time considering that I now have a pretty solid grasp on how to derive sigma notation from a function with equal subdivisions and within a specific interval. It’s a relieving knowing that I’ll have much less trouble working through calculus questions that use this type of notation with Riemann Sums. That being said, I definitely want to make more progress through Integrals this coming week than I did this past week. My goal this week is to get through 7 videos, 1 article, and 4 exercises. Wish me luck!