1. The Area Question and the Integral
Lesson 6.1

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

3. Area Under the Curve
Using more and more rectangles to approximate the area

4. The Area Under a Curve
Divide the area under the curve on the interval [a,b] into n equal segments
Each "rectangle" has height f(xi)
Each width is x
The area if the i th rectangle is f(xi)•x
We sum the areas a b •

5. Summation Notation We use summation notation
Note the basic rules and formulas
Summation Formulas, pg 218

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

7. Practice Summation
Try these

8. 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

9. Limit of a Sum
Now So

10. Limit of a Sum
Continuing …

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

12. The Sum Calculated Consider the function 2x2 – 7x + 5 Use x = 0.1
Let the = left edge of each subinterval
Note the sum

13. The Area Under a Curve
The accuracy of the summation will increase if we have more segments
As we increase n
As n gets infinitely large the summation is exact

14. The Definite Integral
We will use another notation to represent the limit of the summation
Upper limit of integration
Lower limit of integration
The integrand

15. Example
Try
Use summation on calculator.

16. Example
Note increased accuracy with smaller x

17. Limit of the Sum
The definite integral is the limit of the sum.

18. Practice
Try this
What is the summation?
Which gives us
Now take limit

19. Practice Try this one For n = 50? Now take limit What is x?
What is the summation?
For n = 50?
Now take limit

20. Assignment
Lesson 6.1
Page 221
Exercises 1 – odd