Area Approximation This template can be used as a starter file for presenting training materials in a group setting. Sections Right-click on a slide to add sections. Sections can help to organize your slides or facilitate collaboration between multiple authors. Notes Use the Notes section for delivery notes or to provide additional details for the audience. View these notes in Presentation View during your presentation. Keep in mind the font size (important for accessibility, visibility, videotaping, and online production) Coordinated colors Pay particular attention to the graphs, charts, and text boxes. Consider that attendees will print in black and white or grayscale. Run a test print to make sure your colors work when printed in pure black and white and grayscale. Graphics, tables, and graphs Keep it simple: If possible, use consistent, non-distracting styles and colors. Label all graphs and tables. 4-C

Exact Area Use geometric shapes such as rectangles, circles, trapezoids, triangles etc… rectangle parallelogram triangle

Approximate Area Midpoint Trapezoidal Rule

Approximate Area Riemann sums Left endpoint Right endpoint

Inscribed Rectangles: rectangles remain under the curve Inscribed Rectangles: rectangles remain under the curve. Slightly underestimates the area. Circumscribed Rectangles: rectangles are slightly above the curve. Slightly overestimates the area Left Endpoints

Left endpoints: Increasing: inscribed Decreasing: circumscribed Right Endpoints: increasing: circumscribed, decreasing: inscribed

The area under a curve bounded by f(x) and the x-axis and the lines x = a and x = b is given by Where and n is the number of sub-intervals

Therefore: Inscribed rectangles Circumscribed rectangles The sum of the area of the inscribed rectangles is called a lower sum, and the sum of the area of the circumscribed rectangles is called an upper sum http://archives.math.utk.edu/visual.calculus/4/areas.2/index.html

Fundamental Theorem of Calculus: If f(x) is continuous at every point [a, b] and F(x) is an antiderivative of f(x) on [a, b] then the area under the curve can be approximated to be

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Simpson’s Rule:

1) Find the area under the curve from

2) Approximate the area under from With 4 subintervals using inscribed rectangles

3) Approximate the area under from Using the midpoint formula and n = 4

4) Approximate the area under the curve between x = 0 and x = 2 Using the Trapezoidal Rule with 6 subintervals

5) Use Simpson’s Rule to approximate the area under the curve on the interval using 8 subintervals

6) The rectangles used to estimate the area under the curve on the interval using 5 subintervals with right endpoints will be Inscribed Circumscribed Neither both

7) Find the area under the curve on the interval using 4 inscribed rectangles

Home Work Worksheet on Area Use a section header for each of the topics, so there is a clear transition to the audience. Worksheet on Area