1. Area Calculus

2. Area of a Plane Region Calculus was built around two problems
Tangent line
Area

3. Area To approximate area, we use rectangles
More rectangles means more accuracy

4. Area Can over approximate with an upper sum
Or under approximate with a lower sum

5. Area Variables include Number of rectangles used Endpoints used

6. Area
Regardless of the number of rectangles or types of inputs used, the method is basically the same.
Multiply width times height and add.

7. Upper and Lower Sums
An upper sum is defined as the area of circumscribed rectangles
A lower sum is defined as the area of inscribed rectangles
The actual area under a curve is always between these two sums or equal to one or both of them.

8. Area Approximation
We wish to approximate the area under a curve f from a to b.
We begin by subdividing the interval [a, b] into n subintervals.
Each subinterval is of width

9. Area Approximation

10. Area Approximation
We wish to approximate the area under a curve f from a to b.
We begin by subdividing the interval [a, b] into n
subintervals of width
Minimum value of f in the ith subinterval
Maximum value of f in the ith subinterval

11. Area Approximation

12. Area Approximation So the width of each rectangle is

14. Area Approximation So the width of each rectangle is
The height of each rectangle is either or

15. 15

16. Area Approximation So the width of each rectangle is
The height of each rectangle is either or
So the upper and lower sums can be defined as
Lower sum
Upper sum

17. Area Approximation It is important to note that
Neither approximation will give you the actual area
Either approximation can be found to such a degree that it is accurate enough by taking the limit as n goes to infinity
In other words