4.2 Area Greenfield Village, Michigan Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2002

Objectives Use sigma notation to write and evaluate a sum. Understand the concept of area. Approximate the area of a plane region.

Sigma Notation:

Examples:

Properties:

Summation Formulas:

Example:

Example:

We already know how to find the area of some figures. Finding the area of regions other than polygons is more difficult. One way to find the area under a curve is by using rectangles to approximate the area.

Inscribed rectangles are all below the curve: Circumscribed rectangles are all above the curve:

Lower Sum: Inscribed rectangles Example: We could estimate the area under the curve by drawing rectangles touching at their left corners. Approximate area:

Upper Sum: Circumscribed rectangles We could also estimate the area under the curve by drawing rectangles touching at their right corners. Approximate area:

The actual area is between the two. But NOT the average of the two!

Another approach would be to use rectangles that touch at the midpoint. In this example there are four subintervals. As the number of subintervals increases, so does the accuracy. Approximate area:

The exact answer for this problem is . With 8 subintervals: Approximate area: The exact answer for this problem is . width of subinterval

Lower Sum: sum of inscribed rectangles Upper Sum: sum of circumscribed rectangles

To approximate the area of the region: 1. Divide the interval [a,b] into subintervals of length 2. Lower: use the lowest height of each subinterval Upper: use the highest height on each subinterval 3. Add the areas of the rectangles (bh).

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