1. Use sigma notation to write and evaluate a sum Understand the concept of area Approximate the area of a plane region Find the area of a plane region using limits 4.2A Area
2. Last section we looked at antiderivation. This section we will look further into finding area of a region in the plane. These might seem unrelated, but by 4.4 we will find the connection with the Fundamental Theorem of Calculus We will start by looking at a concise notation for sums, called sigma notation.
3. Ex 1. p. 259 Examples of sigma notation Notice with parts a & b that the same sum can be expressed in more than one way.
4. Any variable can be used as the index of summation, but traditionally i, j, and k are used most often. When expanding, the index variable gets replaced and doesn’t appear in the expanded form. Let’s look at developing some rules. A constant can be distributed in or factored out of a sum. You can split up sums and differences
5. Carl Friedrich Gauss (1777-1855) was asked to add all the integers from 1 to 100. In just a few minutes he had the answer, and he had it right. This is what he did: 1 + 2 + 3 + . . . + 100 100+ 99 + 98 + . . . + 1 101 +101+ 101+ . . . + 101 This is shown in Theorem 4.2, rule 2
6. Ex 2 p. 260 Evaluating a sum with the theorem rules Factor constant 1/n 2 out of summation Write as two sums Apply thm 4.2 Simplify a bunch! n 10 0.95 100 0.545 1000 0.5045 10,000 0.50045
7. Notice that as n increases, the sum appears to approach a limit. In other words: n 10 0.95 100 0.545 1000 0.5045 10,000 0.50045
8. In geometry the simplest kind of area is a rectangle. Next would be area of triangle.
10. Ancient Greek mathematicians found area of circle by the exhaustion method! In other words, through a limiting process they squeezed area between two polygons – one inscribed and one circumscribed about the circle.
11. By increasing the number of subdivisions, they got more accuracy. We will do a similar thing involving rectangles with the areas we will look at now. http://youtu.be/OaCVdzr3MjM to see how to estimate areas using rectangles http://youtu.be/mPJ2bTutY14 to put into summation form http://youtu.be/myZSiwxQinw to find exact area under a curve http://youtu.be/Jbn81x30uRs to find exact area with more twists
12. 4.2a p. 267/ 1-6, 7-21 odd and watch these youtube videos one more time on your own!
