1. Warmup 11/16/15 This one is going to be turned in to Mr. C. If you’re not Christian, what would it take for you to become one? If you are one, what converted you to Christianity? To refine our ability to find the area under a curve None Objective Tonight’s Homework

2. Homework Help Let’s spend the first 10 minutes of class going over any problems with which you need help.

3. Notes on Infinite Partitions Under a Curve Let’s look at the area under a curve again.

4. Notes on Infinite Partitions Under a Curve Let’s look at the area under a curve again. As we increase the number of partitions (rectangles), we can see that our answer is going to get more and more accurate.

5. Notes on Infinite Partitions Under a Curve Let’s look at the area under a curve again. As we increase the number of partitions (rectangles), we can see that our answer is going to get more and more accurate. So if we want to find the exact area, all we need to do is use an infinite number of infinitely small rectangles. Limits!

6. Notes on Infinite Partitions Under a Curve How are we going to do this? Well, at simplest, what we want to do is find the area of each rectangle, then add up all the areas for all the rectangles. For now, let’s not worry about infinity.

7. Notes on Infinite Partitions Under a Curve How are we going to do this? Well, at simplest, what we want to do is find the area of each rectangle, then add up all the areas for all the rectangles. For now, let’s not worry about infinity. For some number of rectangles “n”, our total area would be: area 1 + area 2 + area 3 + area 4 + … + area n

8. Notes on Infinite Partitions Under a Curve How do we simplify this? With a summation! Area The above is our quick way of saying “add up the areas of each rectangle from the first rectangle all the way to some rectangle numbered ‘n’”. The area of each rectangle is base times height: BASE each ● HEIGHT each ∑ i=n i=0 ∑ i=n i=0

9. Notes on Infinite Partitions Under a Curve Each of our rectangles is the same width, so base is independent of which rectangle we’re looking at. This means we can pull it out of the summation. BASE each HEIGHT each So how do we get the base of each? Let’s imagine we’re looking at the area under some curve from 0 to 4 with 4 rectangles. Each rectangle would have a base of 1. If we call “0” “a” and “4”, “b” and we divide it up into “n” rectangles, the base of each rectangle is:(b – a) / n ∑ i=n i=0 0=a 4=b

10. Notes on Infinite Partitions Under a Curve Let’s modify our summation with this: (b – a) / n HEIGHT each Now we just have to find a general way to get the height of each rectangle. Where to start? The height of each rectangle is the same as f(x) at the left side of that rectangle. Put in x, get out y. But x changes for each rectangle! We need a formula or something. ∑ i=n i=0 f(x) x

11. Notes on Infinite Partitions Under a Curve For our first rectangle, we need to go over 0 before going up. For our second rectangle, we need to go over 1 rectangle. (or (b-a)/n). For the third, we need to go over 2(b-a)/n. etc. We can generalize this: To get x for rectangle “i”, we need to go over: x i = i(b – a) / n If I put x i into my formula, I’ll get out the y- value or height of the rectangle at that spot.

12. Notes on Infinite Partitions Under a Curve This leaves us with the following: Area = (b – a) / n f(x i ) So what do we have? Let’s summarize: (b – a) / n is the base of each rectangle. That’s the same for every rectangle, so we pulled that out of the summation. f(x i ) is an adding up of all the heights of the rectangles. We start with rectangle 0 and find height. Then we move to the “x” value for rectangle 1 and find height. We keep doing this until we’ve done “n” rectangles. ∑ i=n i=0 ∑ i=n i=0

13. Notes on Infinite Partitions Under a Curve We have just one thing left to mention. We eventually want to add up an infinite number of rectangles, so we need to take the limit of our function as n  ∞. We want the number of rectangles to eventually become infinite. We’ll stop here for today. Right now, you don’t have the tools you need to turn the summation into something we can take a limit of. Just know that this will give us an infinite number of infinitely thin rectangles, which will reduce our error in estimation down to zero. This will give us the perfect area under the curve. ∑ i=n i=0 True Area = lim (b – a) / n f(x i ) n∞n∞

14. Exit Question Which of the following functions wouldn't work well for this method of finding area? a) Linear b) Quadratic c) Logarithmic d) Exponential e) All of the above would work equally well