This week I was given a flood of new information and I feel like I’m barely able to keep it sorted out in my brain. I found everything I learned this week pretty difficult to understand which is why I’m struggling to retain it all. the things I learned in the first few weeks of the unit Integrals seemed pretty straightforward, but everything I went through this was much less intuitive to me. I’m sure that it’s partly because I don’t have a strong grasp on integral notation so I’m sure that it’ll start becoming easier once I get a better feel for the notation. Even though I found this week challenging, I’m pretty confident that the stuff I’m learning will seem relatively simple with a bit more practice. Right now, however, I definitely feel like I’m drowning. 🌊 🆘

The first thing I learned about this week was that it’s possible to find the sum of an integral by splitting it up into two parts and taking the sum of both parts:

This is what’s known as the Sum/Difference property. Initially this concept didn’t seem like it’d be that useful until I learned about the other integral properties and how they could be combined to find the equations for more difficult integrals. The five integral properties I’ve learned about so far are:

My layman’s explanations for these properties are…

• Sum/Difference
  • You can add/subtract two separate integrals, ex. f(x) and g(x), by finding their sum/difference BEFORE finding their integral, ex. _a∫^b[f(x) + g(x)], or by finding their integrals separately and then adding or subtracting the sum/difference afterwards.
  • I believe this means that the interval [a, b] has to be the same for each integral but I’m not sure.
• Constant Multiple
  • If there’s a constant that’s inside of an integral that’s scaling the function, you can pull the constant out of the integral, solve the integral as it is, and then multiply the solution of the integral by the constant afterwards.
• Reverse Interval
  • If you take a standard integral where the interval is being calculated from left to right, conventionally written as [a, b], you can reverse the interval and calculate it by going from right to left, i.e. from [b, a], as long as you change the sign in front of the integral (i.e. if the integral was positive when going from [a, b] then it would be negative when going from [b, a]).
• Adding Integrals
  • This is the property I went over above.
• Zero-length Interval
  • An integral that’s being calculated at a single point, i.e. from [a, a], it equals 0. This makes sense since there would be no width to the integral meaning there would be no area to it, but I don’t understand why this is useful to know.

Once I went over these other integral properties, I then went through a question which showed me how you can find the integral between an interval of [x, x²] by combining the addition and reverse integral properties:

This is a good example of what I was referring to when I said I was struggling with the things I learned this week. I can understand how the solution to this question works when I go through it slowly, but, all in all, it still seems fairly overwhelming to me and I struggle to make sense of the notation. As you’d expect, I’m getting more comfortable with these types of questions the more I work through them so I’m sure I’ll reach a point where they don’t seem nearly as difficult. (And hopefully that’ll happen sooner rather than later.)

One thing that I was really happy about this week was that I think I figured out how/why the first fundamental theory of calculus works:

Another way of thinking about this is:

1. An integral is the area between a function and the x-axis.
2. Slope is a way to measure how much a function is increasing or decreasing between a certain interval.
3. A derivative is the slope of a function at a specific point.
4. The derivative of an integral will be equal to the “height” of the function at that point because the area that’s being added/subtracted to the derivative (a.k.a. how much the integral is increasing or decreasing, i.e. it’s slope) IS the “height” of the function at that point.

I could be wrong, but I’m pretty sure that’s the gist of this theorem. I realized this on Thursday and then spent the next few days working through questions and trying to understand the second fundamental theorem of calculus which, as of now, I don’t understand at all. I have a feeling the second theorem is just as simple as the first but I haven’t been able wrap my head around it yet. Here’s a screen shot from an article on KA that talks about both theorems:

It seems like the second theorem is simply stating, “if you calculate the ‘height’ of the integral at the end points (a.k.a. the left and right bounds) and then subtract one from the other, you’re left with the derivative of the definite integral.” (Honestly, I have no idea what I just wrote but I’m 99% sure whatever I just said is not correct. I really don’t understand what the second theorem is stating or calculating or how/why it works…. 😡)

Even though I don’t understand how the second theorem works, I worked through a few exercises which I think helped me to make a bit of headway on understanding it. Here’s an example question from one of the exercises:

Lastly, on Saturday I briefly learned about what Sal referred to as the Reverse Power Rule. This rule seems pretty straightforward but I haven’t completely figured it out. My understanding is that you use this rule to take an expression out of an antiderivative (which I think is the same thing as an integral) which has a term in it that’s being raised to a power. By taking the expression out of the antiderivative, you’re turning the expression back into a derivative. (Again, I really have no idea if any of what I’m saying is accurate or makes any sense at all…) Here’s a photo from my notes that talks about the rule

There’s more nuance to this rule than that photo explains, but I don’t have a strong enough understanding of it to be able to explain how it works yet. Again, I don’t think this is a concept that will be too difficult to understand once I practice it a bit more but, this on top of my already weak understanding of antiderivatives and integrals in general, right now this rule seems pretty confusing to me.

Writing this post helped me realize how many tricky new concepts I worked through this week. I’m proud of everything I was able to get through even though I didn’t exactly figure much of it out. Nonetheless, I definitely think I’m making progress on understanding integrals which I’m happy about. I’m now 40% of the way through Integrals (1,280/3,200 M.P.) which means, excluding the unit test, I’m halfway through the unit! (The unit test makes up the final 20% of each unit.) This coming week I’m hoping that I’ll be able to figure out a bunch of the concepts I learned this past week. My gut tells me there’ll be a moment where the lightbulbs go off and all of a sudden it all starts to make sense. Hopefully!!! 💡🤞🏼