1. Area as the Limit of a Sum
Lesson 5.2

2. Area Under the Curve
What does the following demo suggest about how to measure the area under the curve?

3. Area under f(x) = ln x
Consider the task to compute the area under a curve f(x) = ln x on interval [1,5] x
We estimate with 4 rectangles using the right endpoints

4. We can improve our estimate by increasing the number of rectangles
Area under the Curve x
We can improve our estimate by increasing the number of rectangles

5. Area under the Curve Increasing the number of rectangles to n
This can be done on the calculator:

6. Generalizing In general … The actual area is where
a b
In general …
The actual area is
where
Try Java Applet Demo

7. Summation Notation We use summation notation
Note the basic rules and formulas
Examples pg. 295
Theorem 5.2 Formulas, pg 296

8. Use of Calculator Note again summation capability of calculator
Syntax is:  (expression, variable, low, high)

9. Practice Summation
Try these

10. Limit of a Sum
a b
For a function f(x), the area under the curve from a to b is where x = (b – a)/n and
Consider the region bounded by f(x) = x2 the axes, and the lines x = 2 and x = 3

11. Limit of a Sum
Now So

12. Limit of a Sum
Continuing …

13. Practice Summation
For our general formula:
let f(x) = 3 – 2x on [0,1]

14. Assignment
Lesson 5.2
Page 303
Exercises 1 – 61 EOO (omit 45)