1. Section 15.3 Area and Definite Integral

2. Area Estimation
How can we estimate the area bounded by the curve y = x2, the lines x = 1 and x = 3, and the x-axis?
We will use 4 rectangles to approximate the area.
Question How can we improve our estimation?

3. Definition of the Area of a Region
The area of the region bounded by the graph of f(x), the x-axis, the vertical lines x = a and x = b is
where ci is the endpoint of the ith interval, and

4. The Definite Integral The definite integral of f from a to b is
where a is called lower limit, b is called upper limit,
If f(x) is above the x-axis, then the definite integral of f (x) from a to b represents the area of the region bounded by the graph of f(x), the x-axis , the vertical lines x = a and x = b.

5. Examples
Evaluate the integrals by interpreting in terms of area.