## Riemann Sums

by Rogelio Yoyontzin
## 1. Lower Sum

## 2. Caluclation of the real Area

### 2.1. We will use the Upper Sums ON Blackboard

2.1.1. We will Calcuate the general formula for any n

2.1.2. We calculate the limit when n is going to infinity to get the area

### 2.2. Play more:

## 3. The Problem

### 3.1. The Graph Step 1: Make a large, careful graph of y = x² - x + 1 on the interval (1,9).

### 3.2. The Area

### 3.3. We want to use 8 rectangles Step 2: Divide the interval (1,9) into n = 8 equal subdivisions, each subinterval having width Δx=(9−1)/8=1, Make a table showing the intervals, their left endpoints xL midpoints xM, and right endpoints xR, and the values of f(x) at each of these points

3.3.1. Upper Sum

3.3.2. Step 3: For each subinterval, draw rectangles with base on the x-axis and height f(xL),f(xM), and f(xR) respectively.

3.3.3. Upper vs Lower

### 3.4. Play with the number of rectangles

### 3.5. The Table we want

### 3.6. Next steps

### 3.7. Final Table

## 4. More Problems