1. Warmup 11/17/15 You’ll turn this one in, just like yesterday’s. If you’re not Christian, what do you think of Christians? Is this different from how you thought before you came to Belleview? If you are Christian, what’s the biggest thing you’ve learned about God this school year? To refine our ability to find the area under a curve pp 228: 2, 3, 6, 7, 9 Objective Tonight’s Homework

2. Notes on Summations of Polynomials Yesterday we found the limit + summation combo to get perfect area under a curve. We want to do an example problem, but first we need to take a detour to learn how to deal with that summation. Copy down the following rules into your cheat sheet. Our book assumes you know them. SUMMATION FORMULAS i 1 = i 2 = i 3 = ∑ i=n i=0 n(n+1) 2 ∑ i=n i=0 n(n+1)(2n+1) 6 n 2 (n+1) 2 4 ∑ i=n i=0

3. Notes on Summations of Polynomials Knowing tricks like this to summation, we can go back to where we left off yesterday. We left off where we were just about to do an example of summing up an infinite number of rectangles. Let’s continue this but with an actual example problem.

4. Notes on Infinite Partitions Under a Curve Example: Find the exact area under y = x 2 from 0 to 3.

5. Notes on Infinite Partitions Under a Curve Example: Find the exact area under y = x 2 from 0 to 3. True Area = lim (b – a) / n f(x i ) Simplify: base = (b-a)/n = (3-0)/n = 3/n For each rectangle, we need to go over x i before going up f(x i ) to get height. So how do I get x i ? The x’s jump by how wide the base is, so I’m going to need that piece, times how many rectangles I’ve jumped over. So: x i = i(base)  x i = i(3/n) ∑ n∞n∞ i=n i=0

6. Notes on Infinite Partitions Under a Curve Our heights, f(x i ), is the “y” we get out when we put in each x i. So we’ll go back to our original equation and replace every x with an x i, then replace each x i with what x i is, to get the special form of our equation that we’ll put into the summation: f(x) = x 2 f(x i ) = x i 2 x i = i(3/n) f(x i ) = [i(3/n)] 2

7. Notes on Infinite Partitions Under a Curve True Area = lim (b – a) / n f(x i ) 3/n [i(3/n)] 2 True Area = lim 3/n [i(3/n)] 2 The only thing we care about in our summation is the stuff with “i”. We can pull the rest out. True Area = lim 3/n (3/n) 2 i 2 ∑ n∞n∞ i=n i=0 ∑ i=n i=0 n∞n∞ ∑ i=n i=0 n∞n∞

8. Notes on Infinite Partitions Under a Curve Look at the summation. It’s one of the special patterns! i 2 This summation is easy. We just replace the summation with the rule we wrote down. = lim (3/n) 3 To finish this, all we have to do is get our “n” terms together in one fraction and we can see what the limit does. Then we just simplify. (n)(n+1)(2n+1) 6 n∞n∞ ∑ i=n i=0

9. Notes on Infinite Partitions Under a Curve = lim (3/n) 3 = lim (n)(n+1)(2n+1) 6 n∞n∞ (n 2 +n)(2n+1) 6 2n 3 + 3n 2 + n 6 n∞n∞ n∞n∞ 27 n 3 2n 3 + 3n 2 + n 6 n∞n∞

10. Notes on Infinite Partitions Under a Curve Next, we can swap denominators (you can do that with multiplication in fractions). = lim 27 6 n∞n∞ 2n 3 + 3n 2 + n n 3

11. Notes on Infinite Partitions Under a Curve Next, we can swap denominators (you can do that with multiplication in fractions). = lim Now let’s take the limit. Remember, look at the highest factors! What happens as n  infinity? This fraction becomes “2/1”. Or just “2”. = 2= = 9 27 6 n∞n∞ 2n 3 + 3n 2 + n n 3 27 6 27 3

12. Group Practice JUST FOR TODAY, DO NOT LOOK AT THE EXAMPLES IN THIS SECTION OF THE BOOK. SAXON DOES A VERY BAD JOB OF EXPLAINING THIS CONCEPT AND IT WILL JUST MAKE YOU CONFUSED IF YOU LOOK AT HIS EXAMPLES. Then look at the homework tonight and see if there are any problems you think will be hard. Now is the time to ask a friend or the teacher for help! pp 228: 2, 3, 6, 7, 9

13. Exit Question What is the area under y = x from 0 to 5? a) 6 b) 8.5 c) 12.5 d) 16 e) 25 f) None of the above