1. AREA
Section 4.2

2. When you are done with your homework, you should be able to…
Use sigma notation to write and evaluate a sum
Understand the concept of area
Approximate the area of a plane region
Find the are of a plane region using limits

3. SIGMA NOTATION The sum of n terms is written as
where i is the index of summation, is the ith term of the sum, and the upper and lower bounds of summation are n and 1.
*The lower bound can be any number less than or equal to the upper bound

4. Evaluate

5. Evaluate

6. Summation Properties 1. 2.

7. Theorem: Summation Formulas1. 2. 3. 4.

8. AREA Recall that the definition of the area of a rectangle is
From this definition, we can develop formulas for many other plane regions

9. AREA continued…
We can approximate the area of f by summing up the areas of the rectangles:
What happens to the area approximation when the width of the rectangles decreases?

10. Theorem: Limits of the Lower and Upper Sums
Let f be continuous and nonnegative on the interval The limits as of both the lower and upper sums exist and are equal to each other. That is,
Where and and are the minimum and maximum values of f on the subinterval.

11. Continued…
Since the same limit is attained for both the minimum value and the maximum, it follows from Squeeze Theorem that the choice of x in the ith subinterval does not affect the limit. This means that we can choose an arbitrary x-value in the ith subinterval.

12. Definition of an Area in the Plane
Let f be continuous and nonnegative on the interval The area of the region bounded by the graph of f, the x-axis, and the vertical lines and is

13. Find the area of the region bounded by the graph , the x-axis, and the vertical lines and .