1. A REA A PPROXIMATION 4-B

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

3. Approximate Area Midpoint Trapezoidal Rule

4. Approximate Area Riemann sums Left endpoint Right endpoint

5. 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

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

7. 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

8. Therefore: Inscribed rectangles Circumscribed rectangles http://archives.math.utk.edu/visual.calculus/4/areas.2/index.html 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

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 a)Inscribed b)Circumscribed c)Neither d)both

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

19. H OME W ORK Worksheet on Area