1. 5.1 Areas and Distances

2. Area Estimation How can we estimate the area bounded by the curve y = x 2, the lines x = 1 and x = 3, and the x -axis? Let’s use 4 rectangles to approximate the area. Question How can we improve our estimation? http://science.kennesaw.edu/~plaval/applets/ Riemann.html

3. Summation Notation The sum of n terms can be written using sigma notation: where i is the index of summation 1 is the lower bound n is the upper bound Example: Evaluate

4. Summation Formulas

5. Example Use summation formulas to simplify.

6. |a|a |b|b A1A1 A2A2 A3A3 A4A4 x1x1 x2x2 x3x3 x4x4 Estimating Areas A ≈ A 1 + A 2 + A 3 + A 4

7. Definition of the Area of a Region The area of the region that lies under the graph of f ( x ) from x = a to x = b is where x i is the endpoint of the i th interval, and

8. Examples Find the area of the region bounded by the graph of f ( x ) = 3 x, the x -axis and the vertical lines x = 1 and x = 6, using a)Geometry Formula. b)The limit Definition of Area.