1. A REA A PPROXIMATION 4-E Riemann Sums

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

3. Approximate Area: Add the area of the Rectangles Midpoint

4. Approximate Area: Add the area of the Rectangles Trapezoidal Rule

5. Approximate Area Riemann sums Left endpoint Right endpoint

6. 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

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

8. The exact 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

9. 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

10. 1) Find the area under the curve from

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

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

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

14. 5) 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

15. 6) Find approximate the area under the curve on the interval using right hand Riemann sum with 4 equal subdivisions

16. 7) Approximate by using 5 rectangles of equal width and an Upper Riemann Sum

17. H OME W ORK Area Approximations worksheet