1. Area and the Definite Integral
Lesson 7.3A

2. 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 •

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

4. 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

5. 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

6. Example
Try
Use summation on calculator.

7. Example
Note increased accuracy with smaller x

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

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

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

11. Assignment
Lesson 7.3A
Page 458
Exercises 6 – 20 all