4.2 Area

Sigma Notation where i is the index of summation, a i is the ith term, and the lower and upper bounds of summation are 1 and n respectively The sum of n terms a 1, a 2, a 3, ….an is written

Examples:

Properties of Summations Summation Formulas

Example 1: Find the sum of the first 100 integers.

Example 2: Summation Practice

Example 3: Limits Review

Example 4: Limit of a Sequence

Warm-up

Definition of the Area of a Rectangle: A=bh Take a rectangle whose area is twice the triangle: A=1/2 bh For any polygon, just divide the polygon into triangles.

Area of Inscribed Polygon < Area Circle < Area of Circumscribed Polygon

Area of a Plane Region Find the area under the curve of Between x = 0 and x = 2

Area of a Plane Region—Upper and Lower Sums Begin by subdividing the interval [a,b] into n subintervals, each of length Endpoints of the subintervals: Because f is continuous, the Extreme Value Theorem guarantees the existence of a min and a max on the interval.

Sum of these areas= lower sum Sum of these areas= upper sum

Example: Find the upper and lower sums for the region bounded by the graph of

Example: Find the area of the region bounded by the graph of