Module 1: Administrative Notes
1. Here are the solutions to the Finding Definite Integrals using Geometry worksheet (link)
2. Take special heed of due dates on the 2 worksheets. The Riemann worksheet is due Monday at class time, and the FTC worksheet is due at the end of class on Monday. I will be able to help you with it then, but you won't be able to finish the whole thing if you wait until Monday.
3. We will do some more exploration of why the FTC works in class next week.
Module 2: Riemann Sum (Review/Practice)
1. Due Monday at the start of class, please complete #922 and #923 relating to Riemann sums. LINK
2. Note that neither problem explicitly tells you whether to use LRAM or RRAM. You will have to make the decision as to which will give you a lower and upper estimate. (Hint: both functions are increasing.)
3. Also note: the intervals of time and position, respectively, are equally spaced. This is not always the case, as we will see in class.
Module 3: Fundamental Theorem of Calculus: First Glimpses (Lesson)
1. Watch the following 4 videos in succession.
| Video A: https://www.youtube.com/watch?v=h6mZLTpTHcU Notes on video A: The main goal here is to become comfortable with the idea of functions defined by integrals. Become comfortable with the notion of "negative area". It is negative in the same sense that negative velocity simply indicates movement in the opposite direction. |
| Video B: https://www.youtube.com/watch?v=C7ducZoLKgw Notes on Video B: What you basically want to take away is that the derivative and the (definite) integral sort of "cancel" each other out, and the x thereby takes the place of all the t's. The variables aren't always x and t. So watch out for what the function is defined by (upper and lower bounds of integration) and what the integral is taken with respect to (the variable after the 'd'). |
| Video C: https://www.youtube.com/watch?v=FcLeaD3UII4 Notes on Video C: The first example is very straightforward if you understood the last video. The second example, however, is more challenging, so pay special attention to it. Basically, when there is is something other than just "x" as your upper bound of integration, sub it in as you normally would AND then multiply it by the derivative of whatever the bound is. |
| Video D: https://www.youtube.com/watch?v=tnoPnyvVjyc Notes on Video D: This is sort of a summary of what we have learned in this lesson so far. Note the switching of bounds of integration. |
1. On this worksheet, due at the end of class on Monday, please complete #924-935 (all), 940-952 even. LINK
The first part is the skill you just learned, and the second part is just evaluating more definite integrals using the FTC part 2 we saw a while back.
If you need extra assistance with some of the first part, consult these videos:
https://www.youtube.com/watch?v=TqGCNNlx6pU
https://www.youtube.com/watch?v=0z52CLjC2C0
The latter video will help with #932. But do try to think about how to tackle it using properties of integrals before "spoiling" how to do it :)