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Area Approximation

Area Approximation. 4-B. Exact Area. Use geometric shapes such as rectangles, circles, trapezoids, triangles etc…. rectangle. parallelogram. triangle. Midpoint Trapezoidal Rule. Approximate Area . Riemann sums Left endpoint Right endpoint . Approximate Area .

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Area Approximation

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  1. Area Approximation 4-B

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

  3. Midpoint • Trapezoidal Rule Approximate Area

  4. Riemann sums • Left endpoint • Right endpoint Approximate Area

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

  6. Leftendpoints: Increasing: inscribed Decreasing: circumscribed RightEndpoints: increasing: circumscribed, decreasing: inscribed

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

  8. 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

  9. 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

  10. + -

  11. Simpson’s Rule:

  12. 1) Find the area under the curve from

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

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

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

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

  17. 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

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

  19. Home Work Worksheet on Area

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